| L(s) = 1 | + 2·7-s + 12·11-s − 4·23-s − 6·25-s − 4·37-s + 40·43-s + 2·49-s + 40·53-s − 40·67-s − 8·71-s + 24·77-s + 3·81-s + 32·107-s − 76·113-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 8·161-s + 163-s + 167-s + 173-s − 12·175-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 3.61·11-s − 0.834·23-s − 6/5·25-s − 0.657·37-s + 6.09·43-s + 2/7·49-s + 5.49·53-s − 4.88·67-s − 0.949·71-s + 2.73·77-s + 1/3·81-s + 3.09·107-s − 7.14·113-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.630·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 0.907·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.840364517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.840364517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 6 T^{2} + 18 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 - 2 T + 2 T^{2} + 26 T^{3} - 62 T^{4} + 26 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 3 | \( 1 - p T^{4} - 92 T^{8} - p^{5} T^{12} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 357 T^{4} + 68228 T^{8} + 357 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( 1 + 69 T^{4} - 53260 T^{8} + 69 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 18 T^{2} + 762 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 2 T + 2 T^{2} + 6 T^{3} - 382 T^{4} + 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 83 T^{2} + 3148 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 18 T^{2} + 1962 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 2 T^{2} + 34 T^{3} + 178 T^{4} + 34 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 106 T^{2} + 6130 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 - 2227 T^{4} + 6943924 T^{8} - 2227 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 59 | \( ( 1 + 178 T^{2} + 14842 T^{4} + 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T^{2} + 6822 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 14144 T^{4} + 103908670 T^{8} + 14144 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 271 T^{2} + 30340 T^{4} - 271 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 8400 T^{4} + 51732158 T^{8} - 8400 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 168 T^{2} + 14862 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 16837 T^{4} + 160234548 T^{8} + 16837 p^{4} T^{12} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.47039968200936507785614265310, −4.44796556877075481354049994206, −4.44123694171491415918942459113, −4.40192871590941943962610578764, −4.14055254073695755221140426155, −4.09447238956979069900645122129, −3.90124336906101845157583574452, −3.83068679895651823353956897380, −3.70129014586588956908359982874, −3.43089054221230323088587337320, −3.36859975231085735097403918149, −3.07232241939097307548279024737, −2.96354749981587596369119171405, −2.86543276238106734483919515053, −2.37376162890877083880523798316, −2.32489601442900194764323464982, −2.23145712745726877292455904214, −2.22313572747283647280687251029, −1.96113408563155824637131164384, −1.50862912661249994078685309106, −1.36884055280420249220681720359, −1.21401235187646321962082220556, −1.05799206138325665008818298850, −0.902931082670838366749231456541, −0.34487683158596834631415735987,
0.34487683158596834631415735987, 0.902931082670838366749231456541, 1.05799206138325665008818298850, 1.21401235187646321962082220556, 1.36884055280420249220681720359, 1.50862912661249994078685309106, 1.96113408563155824637131164384, 2.22313572747283647280687251029, 2.23145712745726877292455904214, 2.32489601442900194764323464982, 2.37376162890877083880523798316, 2.86543276238106734483919515053, 2.96354749981587596369119171405, 3.07232241939097307548279024737, 3.36859975231085735097403918149, 3.43089054221230323088587337320, 3.70129014586588956908359982874, 3.83068679895651823353956897380, 3.90124336906101845157583574452, 4.09447238956979069900645122129, 4.14055254073695755221140426155, 4.40192871590941943962610578764, 4.44123694171491415918942459113, 4.44796556877075481354049994206, 4.47039968200936507785614265310
Plot not available for L-functions of degree greater than 10.
