| L(s) = 1 | + 3·3-s − 27·5-s + 20·7-s + 14·9-s − 67·11-s − 191·13-s − 81·15-s − 52·17-s + 95·19-s + 60·21-s + 35·23-s + 51·25-s − 3·27-s + 33·29-s − 694·31-s − 201·33-s − 540·35-s − 108·37-s − 573·39-s + 54·41-s + 427·43-s − 378·45-s + 79·47-s − 141·49-s − 156·51-s − 1.02e3·53-s + 1.80e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.41·5-s + 1.07·7-s + 0.518·9-s − 1.83·11-s − 4.07·13-s − 1.39·15-s − 0.741·17-s + 1.14·19-s + 0.623·21-s + 0.317·23-s + 0.407·25-s − 0.0213·27-s + 0.211·29-s − 4.02·31-s − 1.06·33-s − 2.60·35-s − 0.479·37-s − 2.35·39-s + 0.205·41-s + 1.51·43-s − 1.25·45-s + 0.245·47-s − 0.411·49-s − 0.428·51-s − 2.65·53-s + 4.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 - p T - 5 T^{2} + 20 p T^{3} + 206 T^{4} - 3778 T^{5} + 206 p^{3} T^{6} + 20 p^{7} T^{7} - 5 p^{9} T^{8} - p^{13} T^{9} + p^{15} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 + 27 T + 678 T^{2} + 11301 T^{3} + 172491 T^{4} + 79252 p^{2} T^{5} + 172491 p^{3} T^{6} + 11301 p^{6} T^{7} + 678 p^{9} T^{8} + 27 p^{12} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 20 T + 541 T^{2} - 10576 T^{3} + 326121 T^{4} - 5096684 T^{5} + 326121 p^{3} T^{6} - 10576 p^{6} T^{7} + 541 p^{9} T^{8} - 20 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 67 T + 5240 T^{2} + 234909 T^{3} + 12355069 T^{4} + 430650384 T^{5} + 12355069 p^{3} T^{6} + 234909 p^{6} T^{7} + 5240 p^{9} T^{8} + 67 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 191 T + 23263 T^{2} + 1975972 T^{3} + 130160100 T^{4} + 6800437850 T^{5} + 130160100 p^{3} T^{6} + 1975972 p^{6} T^{7} + 23263 p^{9} T^{8} + 191 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 52 T + 1049 p T^{2} + 858938 T^{3} + 148584321 T^{4} + 5919760542 T^{5} + 148584321 p^{3} T^{6} + 858938 p^{6} T^{7} + 1049 p^{10} T^{8} + 52 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 35 T + 20207 T^{2} + 978380 T^{3} + 145917734 T^{4} + 28510807182 T^{5} + 145917734 p^{3} T^{6} + 978380 p^{6} T^{7} + 20207 p^{9} T^{8} - 35 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 33 T + 93283 T^{2} - 3143460 T^{3} + 4021617904 T^{4} - 112545301942 T^{5} + 4021617904 p^{3} T^{6} - 3143460 p^{6} T^{7} + 93283 p^{9} T^{8} - 33 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 694 T + 293295 T^{2} + 89030416 T^{3} + 21330547054 T^{4} + 4074301845364 T^{5} + 21330547054 p^{3} T^{6} + 89030416 p^{6} T^{7} + 293295 p^{9} T^{8} + 694 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 108 T + 112997 T^{2} + 15736016 T^{3} + 5870487334 T^{4} + 1040220891912 T^{5} + 5870487334 p^{3} T^{6} + 15736016 p^{6} T^{7} + 112997 p^{9} T^{8} + 108 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 54 T + 133605 T^{2} - 16041144 T^{3} + 13956578754 T^{4} - 1300273544260 T^{5} + 13956578754 p^{3} T^{6} - 16041144 p^{6} T^{7} + 133605 p^{9} T^{8} - 54 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 427 T + 350514 T^{2} - 107229149 T^{3} + 51704815029 T^{4} - 11984059719352 T^{5} + 51704815029 p^{3} T^{6} - 107229149 p^{6} T^{7} + 350514 p^{9} T^{8} - 427 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 79 T + 174226 T^{2} - 35616873 T^{3} + 14145845809 T^{4} - 6598091929200 T^{5} + 14145845809 p^{3} T^{6} - 35616873 p^{6} T^{7} + 174226 p^{9} T^{8} - 79 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 1025 T + 1114001 T^{2} + 12492380 p T^{3} + 387820797018 T^{4} + 151493987499830 T^{5} + 387820797018 p^{3} T^{6} + 12492380 p^{7} T^{7} + 1114001 p^{9} T^{8} + 1025 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 11 p T + 314841 T^{2} - 119639488 T^{3} + 54150244700 T^{4} - 29168168438206 T^{5} + 54150244700 p^{3} T^{6} - 119639488 p^{6} T^{7} + 314841 p^{9} T^{8} - 11 p^{13} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 1413 T + 1501580 T^{2} + 969603223 T^{3} + 570721172847 T^{4} + 264578763799400 T^{5} + 570721172847 p^{3} T^{6} + 969603223 p^{6} T^{7} + 1501580 p^{9} T^{8} + 1413 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 + 915 T + 1067785 T^{2} + 566553200 T^{3} + 459237953412 T^{4} + 199605710669898 T^{5} + 459237953412 p^{3} T^{6} + 566553200 p^{6} T^{7} + 1067785 p^{9} T^{8} + 915 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 1444 T + 1738835 T^{2} - 1285313056 T^{3} + 909827587194 T^{4} - 514925638483704 T^{5} + 909827587194 p^{3} T^{6} - 1285313056 p^{6} T^{7} + 1738835 p^{9} T^{8} - 1444 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 2682 T + 3951879 T^{2} - 4074444804 T^{3} + 3359862126357 T^{4} - 2286571386234630 T^{5} + 3359862126357 p^{3} T^{6} - 4074444804 p^{6} T^{7} + 3951879 p^{9} T^{8} - 2682 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 836 T + 1559675 T^{2} + 533331168 T^{3} + 740734437194 T^{4} + 88399406080312 T^{5} + 740734437194 p^{3} T^{6} + 533331168 p^{6} T^{7} + 1559675 p^{9} T^{8} + 836 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 502 T + 1987727 T^{2} - 467577928 T^{3} + 1670788325010 T^{4} - 215028866842532 T^{5} + 1670788325010 p^{3} T^{6} - 467577928 p^{6} T^{7} + 1987727 p^{9} T^{8} - 502 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 996 T + 1701789 T^{2} - 1680976480 T^{3} + 1891161861930 T^{4} - 1393909514429816 T^{5} + 1891161861930 p^{3} T^{6} - 1680976480 p^{6} T^{7} + 1701789 p^{9} T^{8} - 996 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 424 T + 2608217 T^{2} - 438733712 T^{3} + 3723742747366 T^{4} - 556004805154000 T^{5} + 3723742747366 p^{3} T^{6} - 438733712 p^{6} T^{7} + 2608217 p^{9} T^{8} - 424 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.78160921507630591785315080589, −5.57194763642851550367909909107, −5.28008397526436849544616757966, −5.25664467184542211584070811098, −5.15852983638311556618345754065, −5.00548801607587185353994854069, −4.88761347935602402011867762046, −4.77146758843546211520616448958, −4.42087290318873848810279143645, −4.14005180280633885088456818488, −4.08606120135605189439171040657, −3.82922524355170859501606146299, −3.64451631916979422876961335987, −3.60824609423760511917980158616, −3.36082982094584810487009275217, −2.89349083458740167345388040458, −2.71353586247900507590279722884, −2.57884142974014258804286648350, −2.51505822140080717089746779145, −2.21102959575427282530715688717, −1.87538417874265088063116612744, −1.71178511240147858357702428042, −1.64681525262087360909759758813, −0.985538361848241919883273896039, −0.860384655928935524870333912056, 0, 0, 0, 0, 0,
0.860384655928935524870333912056, 0.985538361848241919883273896039, 1.64681525262087360909759758813, 1.71178511240147858357702428042, 1.87538417874265088063116612744, 2.21102959575427282530715688717, 2.51505822140080717089746779145, 2.57884142974014258804286648350, 2.71353586247900507590279722884, 2.89349083458740167345388040458, 3.36082982094584810487009275217, 3.60824609423760511917980158616, 3.64451631916979422876961335987, 3.82922524355170859501606146299, 4.08606120135605189439171040657, 4.14005180280633885088456818488, 4.42087290318873848810279143645, 4.77146758843546211520616448958, 4.88761347935602402011867762046, 5.00548801607587185353994854069, 5.15852983638311556618345754065, 5.25664467184542211584070811098, 5.28008397526436849544616757966, 5.57194763642851550367909909107, 5.78160921507630591785315080589
