| L(s) = 1 | − 6.87e10·2-s − 3.50e17·3-s + 4.72e21·4-s − 5.44e25·5-s + 2.40e28·6-s − 1.20e31·7-s − 3.24e32·8-s + 5.51e34·9-s + 3.74e36·10-s + 4.97e37·11-s − 1.65e39·12-s − 2.84e40·13-s + 8.31e41·14-s + 1.90e43·15-s + 2.23e43·16-s − 1.37e45·17-s − 3.79e45·18-s + 8.02e45·19-s − 2.57e47·20-s + 4.23e48·21-s − 3.41e48·22-s + 4.09e49·23-s + 1.13e50·24-s + 1.90e51·25-s + 1.95e51·26-s + 4.34e51·27-s − 5.71e52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·3-s + 0.5·4-s − 1.67·5-s + 0.952·6-s − 1.72·7-s − 0.353·8-s + 0.816·9-s + 1.18·10-s + 0.485·11-s − 0.673·12-s − 0.623·13-s + 1.21·14-s + 2.25·15-s + 0.250·16-s − 1.69·17-s − 0.577·18-s + 0.169·19-s − 0.836·20-s + 2.32·21-s − 0.343·22-s + 0.811·23-s + 0.476·24-s + 1.79·25-s + 0.441·26-s + 0.247·27-s − 0.862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(74-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+73/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(37)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{75}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 6.87e10T \) |
| good | 3 | \( 1 + 3.50e17T + 6.75e34T^{2} \) |
| 5 | \( 1 + 5.44e25T + 1.05e51T^{2} \) |
| 7 | \( 1 + 1.20e31T + 4.92e61T^{2} \) |
| 11 | \( 1 - 4.97e37T + 1.05e76T^{2} \) |
| 13 | \( 1 + 2.84e40T + 2.07e81T^{2} \) |
| 17 | \( 1 + 1.37e45T + 6.64e89T^{2} \) |
| 19 | \( 1 - 8.02e45T + 2.23e93T^{2} \) |
| 23 | \( 1 - 4.09e49T + 2.54e99T^{2} \) |
| 29 | \( 1 - 1.93e53T + 5.68e106T^{2} \) |
| 31 | \( 1 - 1.64e54T + 7.40e108T^{2} \) |
| 37 | \( 1 + 2.76e57T + 3.01e114T^{2} \) |
| 41 | \( 1 - 5.83e58T + 5.41e117T^{2} \) |
| 43 | \( 1 - 5.73e59T + 1.75e119T^{2} \) |
| 47 | \( 1 + 1.00e61T + 1.15e122T^{2} \) |
| 53 | \( 1 + 3.54e62T + 7.44e125T^{2} \) |
| 59 | \( 1 + 1.77e64T + 1.87e129T^{2} \) |
| 61 | \( 1 + 8.28e64T + 2.13e130T^{2} \) |
| 67 | \( 1 + 3.24e66T + 2.01e133T^{2} \) |
| 71 | \( 1 + 3.58e67T + 1.38e135T^{2} \) |
| 73 | \( 1 - 1.10e67T + 1.05e136T^{2} \) |
| 79 | \( 1 + 7.67e68T + 3.36e138T^{2} \) |
| 83 | \( 1 + 5.36e69T + 1.23e140T^{2} \) |
| 89 | \( 1 - 2.49e71T + 2.02e142T^{2} \) |
| 97 | \( 1 - 9.96e71T + 1.08e145T^{2} \) |
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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
−12.58805341947394253725274877344, −11.69924772708517400643708234512, −10.59704288706777926072819972010, −8.986834764273305817308949168806, −7.14249160472746145017407124556, −6.39119662768870588145050258849, −4.48555450003534516601196077681, −3.05185698823116177633704619048, −0.63713644765416817125420834111, 0,
0.63713644765416817125420834111, 3.05185698823116177633704619048, 4.48555450003534516601196077681, 6.39119662768870588145050258849, 7.14249160472746145017407124556, 8.986834764273305817308949168806, 10.59704288706777926072819972010, 11.69924772708517400643708234512, 12.58805341947394253725274877344
