| L(s) = 1 | + 4.27e14·2-s − 4.51e29·4-s + 1.38e34·5-s − 6.93e41·7-s − 4.63e44·8-s + 5.90e48·10-s + 5.10e50·11-s − 5.18e54·13-s − 2.96e56·14-s + 8.80e58·16-s − 6.13e60·17-s − 1.21e63·19-s − 6.23e63·20-s + 2.17e65·22-s + 2.67e67·23-s − 1.38e69·25-s − 2.21e69·26-s + 3.13e71·28-s − 2.47e72·29-s + 9.17e73·31-s + 3.31e74·32-s − 2.61e75·34-s − 9.58e75·35-s + 5.23e77·37-s − 5.20e77·38-s − 6.40e78·40-s + 1.38e79·41-s + ⋯ |
| L(s) = 1 | + 0.536·2-s − 0.712·4-s + 0.348·5-s − 1.02·7-s − 0.918·8-s + 0.186·10-s + 0.144·11-s − 0.375·13-s − 0.547·14-s + 0.219·16-s − 0.759·17-s − 0.613·19-s − 0.247·20-s + 0.0773·22-s + 1.04·23-s − 0.878·25-s − 0.201·26-s + 0.726·28-s − 1.01·29-s + 1.38·31-s + 1.03·32-s − 0.407·34-s − 0.355·35-s + 1.23·37-s − 0.329·38-s − 0.319·40-s + 0.204·41-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 | \( 1 - 4.27e14T + 6.33e29T^{2} \) |
| 5 | \( 1 - 1.38e34T + 1.57e69T^{2} \) |
| 7 | \( 1 + 6.93e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 5.10e50T + 1.25e103T^{2} \) |
| 13 | \( 1 + 5.18e54T + 1.90e110T^{2} \) |
| 17 | \( 1 + 6.13e60T + 6.52e121T^{2} \) |
| 19 | \( 1 + 1.21e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 2.67e67T + 6.47e134T^{2} \) |
| 29 | \( 1 + 2.47e72T + 5.98e144T^{2} \) |
| 31 | \( 1 - 9.17e73T + 4.41e147T^{2} \) |
| 37 | \( 1 - 5.23e77T + 1.78e155T^{2} \) |
| 41 | \( 1 - 1.38e79T + 4.63e159T^{2} \) |
| 43 | \( 1 - 6.24e80T + 5.16e161T^{2} \) |
| 47 | \( 1 - 1.29e82T + 3.44e165T^{2} \) |
| 53 | \( 1 + 2.75e85T + 5.05e170T^{2} \) |
| 59 | \( 1 + 8.21e86T + 2.06e175T^{2} \) |
| 61 | \( 1 + 1.57e88T + 5.59e176T^{2} \) |
| 67 | \( 1 + 2.83e87T + 6.04e180T^{2} \) |
| 71 | \( 1 - 3.67e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 1.12e92T + 2.94e184T^{2} \) |
| 79 | \( 1 - 1.47e94T + 7.32e187T^{2} \) |
| 83 | \( 1 - 8.36e94T + 9.74e189T^{2} \) |
| 89 | \( 1 - 2.37e96T + 9.76e192T^{2} \) |
| 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
−9.246411232346587989879279774662, −7.974242730856519995394330381421, −6.59895588715359274623867788138, −5.96197622067070854928192609816, −4.86295566436439537947878007472, −4.04880654252375372063379327771, −3.10518371022692728024840222655, −2.23422928792284768895137566196, −0.814176339640911890778492614171, 0,
0.814176339640911890778492614171, 2.23422928792284768895137566196, 3.10518371022692728024840222655, 4.04880654252375372063379327771, 4.86295566436439537947878007472, 5.96197622067070854928192609816, 6.59895588715359274623867788138, 7.974242730856519995394330381421, 9.246411232346587989879279774662
