| L(s) = 1 | + 8.02e14·2-s + 1.09e28·4-s + 7.58e34·5-s − 6.40e41·7-s − 5.00e44·8-s + 6.09e49·10-s + 5.91e51·11-s − 1.66e55·13-s − 5.14e56·14-s − 4.08e59·16-s − 9.01e60·17-s − 3.56e61·19-s + 8.29e62·20-s + 4.74e66·22-s − 4.49e67·23-s + 4.18e69·25-s − 1.33e70·26-s − 7.00e69·28-s + 2.30e71·29-s + 1.57e72·31-s − 1.10e73·32-s − 7.24e75·34-s − 4.86e76·35-s − 1.15e76·37-s − 2.85e76·38-s − 3.79e79·40-s − 4.61e79·41-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.0172·4-s + 1.91·5-s − 0.942·7-s − 0.991·8-s + 1.92·10-s + 1.67·11-s − 1.20·13-s − 0.950·14-s − 1.01·16-s − 1.11·17-s − 0.0179·19-s + 0.0329·20-s + 1.68·22-s − 1.76·23-s + 2.65·25-s − 1.21·26-s − 0.0162·28-s + 0.0940·29-s + 0.0237·31-s − 0.0344·32-s − 1.12·34-s − 1.80·35-s − 0.0272·37-s − 0.0180·38-s − 1.89·40-s − 0.678·41-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
| \(L(50)\) |
\(\approx\) |
\(4.080132824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.080132824\) |
| \(L(\frac{101}{2})\) |
|
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 | \( 1 - 8.02e14T + 6.33e29T^{2} \) |
| 5 | \( 1 - 7.58e34T + 1.57e69T^{2} \) |
| 7 | \( 1 + 6.40e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 5.91e51T + 1.25e103T^{2} \) |
| 13 | \( 1 + 1.66e55T + 1.90e110T^{2} \) |
| 17 | \( 1 + 9.01e60T + 6.52e121T^{2} \) |
| 19 | \( 1 + 3.56e61T + 3.95e126T^{2} \) |
| 23 | \( 1 + 4.49e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 2.30e71T + 5.98e144T^{2} \) |
| 31 | \( 1 - 1.57e72T + 4.41e147T^{2} \) |
| 37 | \( 1 + 1.15e76T + 1.78e155T^{2} \) |
| 41 | \( 1 + 4.61e79T + 4.63e159T^{2} \) |
| 43 | \( 1 - 6.68e80T + 5.16e161T^{2} \) |
| 47 | \( 1 + 6.01e82T + 3.44e165T^{2} \) |
| 53 | \( 1 - 2.20e85T + 5.05e170T^{2} \) |
| 59 | \( 1 - 2.04e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 4.09e88T + 5.59e176T^{2} \) |
| 67 | \( 1 - 5.53e89T + 6.04e180T^{2} \) |
| 71 | \( 1 - 4.60e89T + 1.88e183T^{2} \) |
| 73 | \( 1 + 1.94e92T + 2.94e184T^{2} \) |
| 79 | \( 1 - 1.15e94T + 7.32e187T^{2} \) |
| 83 | \( 1 - 1.43e95T + 9.74e189T^{2} \) |
| 89 | \( 1 - 5.96e95T + 9.76e192T^{2} \) |
| 97 | \( 1 - 3.52e98T + 4.90e196T^{2} \) |
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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
−9.526352031859880975677677011705, −8.848652377953305377953647087780, −6.68993198146958167098586601599, −6.36206180526285963718569329965, −5.50401684544324785528541714278, −4.52754067080385968876393743301, −3.59470362174268540289491779461, −2.47292844483358430197366874191, −1.89677823738907723018967753058, −0.56368647620856740001517602400,
0.56368647620856740001517602400, 1.89677823738907723018967753058, 2.47292844483358430197366874191, 3.59470362174268540289491779461, 4.52754067080385968876393743301, 5.50401684544324785528541714278, 6.36206180526285963718569329965, 6.68993198146958167098586601599, 8.848652377953305377953647087780, 9.526352031859880975677677011705
