| L(s) = 1 | − 4·2-s + 12·4-s − 10·5-s − 2·7-s − 32·8-s + 40·10-s + 48·11-s − 80·13-s + 8·14-s + 80·16-s + 108·17-s − 188·19-s − 120·20-s − 192·22-s + 138·23-s + 75·25-s + 320·26-s − 24·28-s + 30·29-s − 188·31-s − 192·32-s − 432·34-s + 20·35-s − 332·37-s + 752·38-s + 320·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.107·7-s − 1.41·8-s + 1.26·10-s + 1.31·11-s − 1.70·13-s + 0.152·14-s + 5/4·16-s + 1.54·17-s − 2.27·19-s − 1.34·20-s − 1.86·22-s + 1.25·23-s + 3/5·25-s + 2.41·26-s − 0.161·28-s + 0.192·29-s − 1.08·31-s − 1.06·32-s − 2.17·34-s + 0.0965·35-s − 1.47·37-s + 3.21·38-s + 1.26·40-s + 0.0228·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9745216805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9745216805\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 393 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 48 T + 1702 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 5610 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 108 T + 12358 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 188 T + 18498 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 p T + 19009 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 30 T + 42067 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 188 T + 55722 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 332 T + 97758 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 42595 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 556 T + 233898 T^{2} - 556 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 162 T - 4679 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 684 T + 414622 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 228 T + 178930 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 254 T + 300747 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 826 T + 515001 T^{2} - 826 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 912 T + 729358 T^{2} - 912 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 232 T - 38814 T^{2} - 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 476 T + 480906 T^{2} + 476 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 306 T - 108143 T^{2} + 306 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1086 T + 1703923 T^{2} + 1086 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 868 T + 809478 T^{2} - 868 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05413999147347269551337213788, −9.564562962529977199466240535237, −9.171976425416285500659916806392, −8.886041490824291130424599138939, −8.332103104416817207571325740054, −8.125671649431090761410923690060, −7.39959813110952492549476482260, −7.27952412616309344540967901622, −6.69552714373068232258278085986, −6.60979281421221555057929167452, −5.63168887873461483684598337838, −5.42943906097906230676548690347, −4.42005980904634625167323855948, −4.30556781913092101867738213853, −3.29552977730500513395626473570, −3.20679448475344968900966212138, −2.13652171562494744656955914936, −1.87084129128829390012299754556, −0.823901913047873236972572815942, −0.45702057764501430077563242542,
0.45702057764501430077563242542, 0.823901913047873236972572815942, 1.87084129128829390012299754556, 2.13652171562494744656955914936, 3.20679448475344968900966212138, 3.29552977730500513395626473570, 4.30556781913092101867738213853, 4.42005980904634625167323855948, 5.42943906097906230676548690347, 5.63168887873461483684598337838, 6.60979281421221555057929167452, 6.69552714373068232258278085986, 7.27952412616309344540967901622, 7.39959813110952492549476482260, 8.125671649431090761410923690060, 8.332103104416817207571325740054, 8.886041490824291130424599138939, 9.171976425416285500659916806392, 9.564562962529977199466240535237, 10.05413999147347269551337213788
