| L(s) = 1 | − 3.44e13·2-s − 6.32e29·4-s − 2.50e34·5-s + 1.15e42·7-s + 4.36e43·8-s + 8.62e47·10-s − 4.43e51·11-s − 7.70e54·13-s − 3.98e55·14-s + 3.99e59·16-s + 1.02e61·17-s − 1.80e63·19-s + 1.58e64·20-s + 1.52e65·22-s + 1.87e67·23-s − 9.51e68·25-s + 2.65e68·26-s − 7.30e71·28-s + 3.52e72·29-s − 2.47e73·31-s − 4.14e73·32-s − 3.54e74·34-s − 2.88e76·35-s − 7.46e77·37-s + 6.22e76·38-s − 1.09e78·40-s + 1.04e79·41-s + ⋯ |
| L(s) = 1 | − 0.0433·2-s − 0.998·4-s − 0.629·5-s + 1.69·7-s + 0.0865·8-s + 0.0272·10-s − 1.25·11-s − 0.558·13-s − 0.0735·14-s + 0.994·16-s + 1.27·17-s − 0.908·19-s + 0.628·20-s + 0.0542·22-s + 0.735·23-s − 0.603·25-s + 0.0241·26-s − 1.69·28-s + 1.43·29-s − 0.371·31-s − 0.129·32-s − 0.0550·34-s − 1.06·35-s − 1.76·37-s + 0.0393·38-s − 0.0545·40-s + 0.153·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(100-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+99/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(50)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 3.44e13T + 6.33e29T^{2} \) |
| 5 | \( 1 + 2.50e34T + 1.57e69T^{2} \) |
| 7 | \( 1 - 1.15e42T + 4.62e83T^{2} \) |
| 11 | \( 1 + 4.43e51T + 1.25e103T^{2} \) |
| 13 | \( 1 + 7.70e54T + 1.90e110T^{2} \) |
| 17 | \( 1 - 1.02e61T + 6.52e121T^{2} \) |
| 19 | \( 1 + 1.80e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 1.87e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 3.52e72T + 5.98e144T^{2} \) |
| 31 | \( 1 + 2.47e73T + 4.41e147T^{2} \) |
| 37 | \( 1 + 7.46e77T + 1.78e155T^{2} \) |
| 41 | \( 1 - 1.04e79T + 4.63e159T^{2} \) |
| 43 | \( 1 - 7.86e80T + 5.16e161T^{2} \) |
| 47 | \( 1 + 8.16e82T + 3.44e165T^{2} \) |
| 53 | \( 1 + 7.29e83T + 5.05e170T^{2} \) |
| 59 | \( 1 - 4.75e87T + 2.06e175T^{2} \) |
| 61 | \( 1 + 1.27e88T + 5.59e176T^{2} \) |
| 67 | \( 1 + 3.93e89T + 6.04e180T^{2} \) |
| 71 | \( 1 - 7.82e91T + 1.88e183T^{2} \) |
| 73 | \( 1 + 1.50e92T + 2.94e184T^{2} \) |
| 79 | \( 1 + 8.21e93T + 7.32e187T^{2} \) |
| 83 | \( 1 + 9.12e94T + 9.74e189T^{2} \) |
| 89 | \( 1 + 4.19e96T + 9.76e192T^{2} \) |
| 97 | \( 1 - 1.11e98T + 4.90e196T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.572008403029106920551659326848, −8.085866850400536276029417121060, −7.35273806646044644811873386901, −5.49871984928479896048523871075, −4.93874472208229611143437321243, −4.21662574458761884254935541818, −3.06832107583813225556224186820, −1.86775645041334747719264467133, −0.873607656147474100538000385052, 0,
0.873607656147474100538000385052, 1.86775645041334747719264467133, 3.06832107583813225556224186820, 4.21662574458761884254935541818, 4.93874472208229611143437321243, 5.49871984928479896048523871075, 7.35273806646044644811873386901, 8.085866850400536276029417121060, 8.572008403029106920551659326848
