| L(s) = 1 | − 8·2-s + 12·3-s + 40·4-s + 20·5-s − 96·6-s − 7-s − 160·8-s + 90·9-s − 160·10-s + 73·11-s + 480·12-s + 68·13-s + 8·14-s + 240·15-s + 560·16-s + 54·17-s − 720·18-s + 79·19-s + 800·20-s − 12·21-s − 584·22-s + 39·23-s − 1.92e3·24-s + 250·25-s − 544·26-s + 540·27-s − 40·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s − 6.53·6-s − 0.0539·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s + 2.00·11-s + 11.5·12-s + 1.45·13-s + 0.152·14-s + 4.13·15-s + 35/4·16-s + 0.770·17-s − 9.42·18-s + 0.953·19-s + 8.94·20-s − 0.124·21-s − 5.65·22-s + 0.353·23-s − 16.3·24-s + 2·25-s − 4.10·26-s + 3.84·27-s − 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(22.35875829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.35875829\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 31 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + T + 566 T^{2} + 285 p T^{3} + 288626 T^{4} + 285 p^{4} T^{5} + 566 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 73 T + 596 p T^{2} - 286409 T^{3} + 13836950 T^{4} - 286409 p^{3} T^{5} + 596 p^{7} T^{6} - 73 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 68 T + 2514 T^{2} - 11620 p T^{3} + 10178986 T^{4} - 11620 p^{4} T^{5} + 2514 p^{6} T^{6} - 68 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 54 T + 524 p T^{2} - 577170 T^{3} + 46635350 T^{4} - 577170 p^{3} T^{5} + 524 p^{7} T^{6} - 54 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 79 T + 21368 T^{2} - 1488519 T^{3} + 10782026 p T^{4} - 1488519 p^{3} T^{5} + 21368 p^{6} T^{6} - 79 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 39 T + 35712 T^{2} - 1592079 T^{3} + 581433230 T^{4} - 1592079 p^{3} T^{5} + 35712 p^{6} T^{6} - 39 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 6 T + 96210 T^{2} + 425238 T^{3} + 3503484626 T^{4} + 425238 p^{3} T^{5} + 96210 p^{6} T^{6} + 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 202 T + 116950 T^{2} - 19964330 T^{3} + 6491226074 T^{4} - 19964330 p^{3} T^{5} + 116950 p^{6} T^{6} - 202 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 162 T + 6256 p T^{2} - 31114206 T^{3} + 25974232350 T^{4} - 31114206 p^{3} T^{5} + 6256 p^{7} T^{6} - 162 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 821 T + 482152 T^{2} - 199093501 T^{3} + 64036145630 T^{4} - 199093501 p^{3} T^{5} + 482152 p^{6} T^{6} - 821 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 750 T + 524264 T^{2} - 228198078 T^{3} + 86067907566 T^{4} - 228198078 p^{3} T^{5} + 524264 p^{6} T^{6} - 750 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 397 T + 163574 T^{2} + 31813981 T^{3} - 5826365542 T^{4} + 31813981 p^{3} T^{5} + 163574 p^{6} T^{6} - 397 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 522 T + 601118 T^{2} + 322908574 T^{3} + 164963394162 T^{4} + 322908574 p^{3} T^{5} + 601118 p^{6} T^{6} + 522 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 384 T + 343732 T^{2} + 96182368 T^{3} - 852620538 T^{4} + 96182368 p^{3} T^{5} + 343732 p^{6} T^{6} - 384 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 58 T - 106022 T^{2} - 122744494 T^{3} + 94807914778 T^{4} - 122744494 p^{3} T^{5} - 106022 p^{6} T^{6} - 58 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 117 T + 504614 T^{2} + 114017211 T^{3} + 178048666866 T^{4} + 114017211 p^{3} T^{5} + 504614 p^{6} T^{6} + 117 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 1361 T + 1709136 T^{2} - 1366562053 T^{3} + 1022740880530 T^{4} - 1366562053 p^{3} T^{5} + 1709136 p^{6} T^{6} - 1361 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 1443 T + 2315364 T^{2} - 1939964859 T^{3} + 1750299266902 T^{4} - 1939964859 p^{3} T^{5} + 2315364 p^{6} T^{6} - 1443 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 220 T + 1611232 T^{2} + 612201788 T^{3} + 1164952426382 T^{4} + 612201788 p^{3} T^{5} + 1611232 p^{6} T^{6} + 220 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 357 T + 873632 T^{2} + 821052477 T^{3} + 648663988602 T^{4} + 821052477 p^{3} T^{5} + 873632 p^{6} T^{6} + 357 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 384 T + 1332412 T^{2} - 461206912 T^{3} + 962649370374 T^{4} - 461206912 p^{3} T^{5} + 1332412 p^{6} T^{6} + 384 p^{9} T^{7} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.06565879005271416670465649655, −6.58515898331001620446724600322, −6.44129893791567908113636766297, −6.38342423514925138569353373006, −6.35065798083373572801723535990, −5.68904071970219410061116034685, −5.42696285511536947018322447044, −5.42608118225146330648530872568, −5.42473486540660145536307436830, −4.24581062006846717329583355085, −4.24463283544361437234812076711, −4.22268461330202292135041485251, −3.88451460944592946803432501966, −3.27568596958708063069399003963, −3.07346100861656945023676751301, −3.05795893289063378419410603554, −2.89349050658027393747650299492, −2.22462440950124216320454616197, −1.99269192657884812719661217566, −1.87696655892322305157286575629, −1.84434968445066371369127251791, −1.07306421896503742320019902481, −1.04473252293446430463408504809, −0.825898386437800311277164346434, −0.72649995555610872365497946158,
0.72649995555610872365497946158, 0.825898386437800311277164346434, 1.04473252293446430463408504809, 1.07306421896503742320019902481, 1.84434968445066371369127251791, 1.87696655892322305157286575629, 1.99269192657884812719661217566, 2.22462440950124216320454616197, 2.89349050658027393747650299492, 3.05795893289063378419410603554, 3.07346100861656945023676751301, 3.27568596958708063069399003963, 3.88451460944592946803432501966, 4.22268461330202292135041485251, 4.24463283544361437234812076711, 4.24581062006846717329583355085, 5.42473486540660145536307436830, 5.42608118225146330648530872568, 5.42696285511536947018322447044, 5.68904071970219410061116034685, 6.35065798083373572801723535990, 6.38342423514925138569353373006, 6.44129893791567908113636766297, 6.58515898331001620446724600322, 7.06565879005271416670465649655
