| L(s) = 1 | − 1.52e14·2-s − 6.10e29·4-s − 2.91e34·5-s + 1.63e41·7-s + 1.90e44·8-s + 4.45e48·10-s − 4.07e51·11-s − 3.82e54·13-s − 2.49e55·14-s + 3.57e59·16-s − 6.53e60·17-s + 2.86e63·19-s + 1.78e64·20-s + 6.21e65·22-s + 8.57e65·23-s − 7.25e68·25-s + 5.84e68·26-s − 9.97e70·28-s − 3.12e72·29-s − 4.13e73·31-s − 1.75e74·32-s + 9.98e74·34-s − 4.77e75·35-s + 4.41e77·37-s − 4.37e77·38-s − 5.54e78·40-s − 8.37e79·41-s + ⋯ |
| L(s) = 1 | − 0.191·2-s − 0.963·4-s − 0.734·5-s + 0.240·7-s + 0.376·8-s + 0.140·10-s − 1.15·11-s − 0.277·13-s − 0.0461·14-s + 0.890·16-s − 0.809·17-s + 1.44·19-s + 0.707·20-s + 0.220·22-s + 0.0336·23-s − 0.460·25-s + 0.0531·26-s − 0.231·28-s − 1.27·29-s − 0.621·31-s − 0.547·32-s + 0.155·34-s − 0.176·35-s + 1.04·37-s − 0.276·38-s − 0.276·40-s − 1.23·41-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 | \( 1 + 1.52e14T + 6.33e29T^{2} \) |
| 5 | \( 1 + 2.91e34T + 1.57e69T^{2} \) |
| 7 | \( 1 - 1.63e41T + 4.62e83T^{2} \) |
| 11 | \( 1 + 4.07e51T + 1.25e103T^{2} \) |
| 13 | \( 1 + 3.82e54T + 1.90e110T^{2} \) |
| 17 | \( 1 + 6.53e60T + 6.52e121T^{2} \) |
| 19 | \( 1 - 2.86e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 8.57e65T + 6.47e134T^{2} \) |
| 29 | \( 1 + 3.12e72T + 5.98e144T^{2} \) |
| 31 | \( 1 + 4.13e73T + 4.41e147T^{2} \) |
| 37 | \( 1 - 4.41e77T + 1.78e155T^{2} \) |
| 41 | \( 1 + 8.37e79T + 4.63e159T^{2} \) |
| 43 | \( 1 - 1.81e80T + 5.16e161T^{2} \) |
| 47 | \( 1 - 7.73e82T + 3.44e165T^{2} \) |
| 53 | \( 1 - 2.30e85T + 5.05e170T^{2} \) |
| 59 | \( 1 + 1.71e87T + 2.06e175T^{2} \) |
| 61 | \( 1 + 5.17e86T + 5.59e176T^{2} \) |
| 67 | \( 1 + 4.08e90T + 6.04e180T^{2} \) |
| 71 | \( 1 + 3.60e90T + 1.88e183T^{2} \) |
| 73 | \( 1 - 2.35e92T + 2.94e184T^{2} \) |
| 79 | \( 1 + 1.88e93T + 7.32e187T^{2} \) |
| 83 | \( 1 + 1.11e94T + 9.74e189T^{2} \) |
| 89 | \( 1 - 1.62e96T + 9.76e192T^{2} \) |
| 97 | \( 1 - 2.22e98T + 4.90e196T^{2} \) |
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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
−8.908552924484972631657790843582, −7.85280313395521187115541239639, −7.38641306640572924712018634587, −5.67681637528115430458050512359, −4.92776811001018723597783257284, −4.03797631529496897182483103386, −3.12979525680152022957263105040, −1.91277815829284603817581335025, −0.69054581134070049711089598120, 0,
0.69054581134070049711089598120, 1.91277815829284603817581335025, 3.12979525680152022957263105040, 4.03797631529496897182483103386, 4.92776811001018723597783257284, 5.67681637528115430458050512359, 7.38641306640572924712018634587, 7.85280313395521187115541239639, 8.908552924484972631657790843582
