| L(s) = 1 | − 1.43e15·2-s + 1.41e30·4-s − 7.34e34·5-s + 3.58e41·7-s − 1.12e45·8-s + 1.05e50·10-s − 1.06e51·11-s + 1.43e55·13-s − 5.13e56·14-s + 7.11e59·16-s + 1.15e61·17-s + 2.87e63·19-s − 1.04e65·20-s + 1.52e66·22-s − 2.29e67·23-s + 3.81e69·25-s − 2.05e70·26-s + 5.08e71·28-s + 4.51e72·29-s + 6.44e73·31-s − 3.06e74·32-s − 1.64e76·34-s − 2.63e76·35-s − 3.44e77·37-s − 4.11e78·38-s + 8.26e79·40-s − 6.94e79·41-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 2.23·4-s − 1.84·5-s + 0.527·7-s − 2.22·8-s + 3.32·10-s − 0.299·11-s + 1.03·13-s − 0.949·14-s + 1.77·16-s + 1.42·17-s + 1.44·19-s − 4.13·20-s + 0.539·22-s − 0.901·23-s + 2.42·25-s − 1.87·26-s + 1.18·28-s + 1.84·29-s + 0.969·31-s − 0.959·32-s − 2.56·34-s − 0.975·35-s − 0.814·37-s − 2.60·38-s + 4.12·40-s − 1.01·41-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 | \( 1 + 1.43e15T + 6.33e29T^{2} \) |
| 5 | \( 1 + 7.34e34T + 1.57e69T^{2} \) |
| 7 | \( 1 - 3.58e41T + 4.62e83T^{2} \) |
| 11 | \( 1 + 1.06e51T + 1.25e103T^{2} \) |
| 13 | \( 1 - 1.43e55T + 1.90e110T^{2} \) |
| 17 | \( 1 - 1.15e61T + 6.52e121T^{2} \) |
| 19 | \( 1 - 2.87e63T + 3.95e126T^{2} \) |
| 23 | \( 1 + 2.29e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 4.51e72T + 5.98e144T^{2} \) |
| 31 | \( 1 - 6.44e73T + 4.41e147T^{2} \) |
| 37 | \( 1 + 3.44e77T + 1.78e155T^{2} \) |
| 41 | \( 1 + 6.94e79T + 4.63e159T^{2} \) |
| 43 | \( 1 + 3.90e80T + 5.16e161T^{2} \) |
| 47 | \( 1 - 6.43e81T + 3.44e165T^{2} \) |
| 53 | \( 1 - 1.75e84T + 5.05e170T^{2} \) |
| 59 | \( 1 + 8.45e86T + 2.06e175T^{2} \) |
| 61 | \( 1 + 2.92e88T + 5.59e176T^{2} \) |
| 67 | \( 1 - 1.96e90T + 6.04e180T^{2} \) |
| 71 | \( 1 - 6.52e91T + 1.88e183T^{2} \) |
| 73 | \( 1 + 6.80e91T + 2.94e184T^{2} \) |
| 79 | \( 1 + 4.81e93T + 7.32e187T^{2} \) |
| 83 | \( 1 - 7.54e94T + 9.74e189T^{2} \) |
| 89 | \( 1 + 3.48e96T + 9.76e192T^{2} \) |
| 97 | \( 1 - 1.55e98T + 4.90e196T^{2} \) |
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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
−8.526448298554868059176114828789, −8.096046245301699782061347757556, −7.52481624738205054169762896067, −6.49466953154555796854891446848, −4.98109149638675169334365191308, −3.61578644923572520112179710101, −2.90978221525225601059517893632, −1.35507652758291544189821395757, −0.901226219545242549287349586998, 0,
0.901226219545242549287349586998, 1.35507652758291544189821395757, 2.90978221525225601059517893632, 3.61578644923572520112179710101, 4.98109149638675169334365191308, 6.49466953154555796854891446848, 7.52481624738205054169762896067, 8.096046245301699782061347757556, 8.526448298554868059176114828789
