| L(s) = 1 | − 2.62e5·2-s − 7.39e8·3-s + 6.87e10·4-s − 3.47e12·5-s + 1.93e14·6-s + 6.42e15·7-s − 1.80e16·8-s + 9.60e16·9-s + 9.12e17·10-s + 2.60e19·11-s − 5.07e19·12-s − 1.70e20·13-s − 1.68e21·14-s + 2.57e21·15-s + 4.72e21·16-s − 6.89e22·17-s − 2.51e22·18-s − 6.71e23·19-s − 2.39e23·20-s − 4.75e24·21-s − 6.82e24·22-s + 1.55e25·23-s + 1.33e25·24-s − 6.06e25·25-s + 4.47e25·26-s + 2.61e26·27-s + 4.41e26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s − 0.407·5-s + 0.778·6-s + 1.49·7-s − 0.353·8-s + 0.213·9-s + 0.288·10-s + 1.41·11-s − 0.550·12-s − 0.420·13-s − 1.05·14-s + 0.449·15-s + 0.250·16-s − 1.18·17-s − 0.150·18-s − 1.48·19-s − 0.203·20-s − 1.64·21-s − 0.997·22-s + 0.998·23-s + 0.389·24-s − 0.833·25-s + 0.297·26-s + 0.866·27-s + 0.745·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.62e5T \) |
| good | 3 | \( 1 + 7.39e8T + 4.50e17T^{2} \) |
| 5 | \( 1 + 3.47e12T + 7.27e25T^{2} \) |
| 7 | \( 1 - 6.42e15T + 1.85e31T^{2} \) |
| 11 | \( 1 - 2.60e19T + 3.40e38T^{2} \) |
| 13 | \( 1 + 1.70e20T + 1.64e41T^{2} \) |
| 17 | \( 1 + 6.89e22T + 3.36e45T^{2} \) |
| 19 | \( 1 + 6.71e23T + 2.06e47T^{2} \) |
| 23 | \( 1 - 1.55e25T + 2.42e50T^{2} \) |
| 29 | \( 1 - 1.41e27T + 1.28e54T^{2} \) |
| 31 | \( 1 + 3.59e27T + 1.51e55T^{2} \) |
| 37 | \( 1 + 3.04e28T + 1.05e58T^{2} \) |
| 41 | \( 1 + 7.95e28T + 4.70e59T^{2} \) |
| 43 | \( 1 + 2.87e30T + 2.74e60T^{2} \) |
| 47 | \( 1 + 1.27e31T + 7.37e61T^{2} \) |
| 53 | \( 1 - 1.70e31T + 6.28e63T^{2} \) |
| 59 | \( 1 + 7.46e32T + 3.32e65T^{2} \) |
| 61 | \( 1 + 2.64e32T + 1.14e66T^{2} \) |
| 67 | \( 1 - 1.82e33T + 3.67e67T^{2} \) |
| 71 | \( 1 + 7.50e33T + 3.13e68T^{2} \) |
| 73 | \( 1 + 4.19e34T + 8.76e68T^{2} \) |
| 79 | \( 1 - 1.22e35T + 1.63e70T^{2} \) |
| 83 | \( 1 + 7.58e34T + 1.01e71T^{2} \) |
| 89 | \( 1 - 2.32e35T + 1.34e72T^{2} \) |
| 97 | \( 1 + 8.69e36T + 3.24e73T^{2} \) |
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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
−17.74585028721314931211162638721, −16.95442136161864325179646046870, −14.86115628879752871306627753342, −11.82610032238276509984628637129, −10.96672369651531591203545136158, −8.570977731502920634196996526093, −6.62689572714781378280534232463, −4.64627168930423818707515840232, −1.60438840656124647413088528658, 0,
1.60438840656124647413088528658, 4.64627168930423818707515840232, 6.62689572714781378280534232463, 8.570977731502920634196996526093, 10.96672369651531591203545136158, 11.82610032238276509984628637129, 14.86115628879752871306627753342, 16.95442136161864325179646046870, 17.74585028721314931211162638721
