| L(s) = 1 | + 2.67e10·2-s − 1.17e16·3-s − 1.64e21·4-s + 8.36e24·5-s − 3.14e26·6-s + 1.48e30·7-s − 1.07e32·8-s − 7.37e33·9-s + 2.24e35·10-s − 3.47e36·11-s + 1.93e37·12-s − 1.64e39·13-s + 3.98e40·14-s − 9.83e40·15-s + 1.00e42·16-s + 5.97e43·17-s − 1.97e44·18-s + 3.55e45·19-s − 1.37e46·20-s − 1.74e46·21-s − 9.30e46·22-s − 5.06e46·23-s + 1.26e48·24-s + 2.76e49·25-s − 4.40e49·26-s + 1.74e50·27-s − 2.44e51·28-s + ⋯ |
| L(s) = 1 | + 0.551·2-s − 0.135·3-s − 0.696·4-s + 1.28·5-s − 0.0747·6-s + 1.48·7-s − 0.935·8-s − 0.981·9-s + 0.708·10-s − 0.372·11-s + 0.0943·12-s − 0.468·13-s + 0.818·14-s − 0.174·15-s + 0.180·16-s + 1.24·17-s − 0.541·18-s + 1.42·19-s − 0.894·20-s − 0.201·21-s − 0.205·22-s − 0.0230·23-s + 0.126·24-s + 0.652·25-s − 0.258·26-s + 0.268·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(72-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+71/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(36)\) |
\(\approx\) |
\(2.906560274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.906560274\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.67e10T + 2.36e21T^{2} \) |
| 3 | \( 1 + 1.17e16T + 7.50e33T^{2} \) |
| 5 | \( 1 - 8.36e24T + 4.23e49T^{2} \) |
| 7 | \( 1 - 1.48e30T + 1.00e60T^{2} \) |
| 11 | \( 1 + 3.47e36T + 8.68e73T^{2} \) |
| 13 | \( 1 + 1.64e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 5.97e43T + 2.30e87T^{2} \) |
| 19 | \( 1 - 3.55e45T + 6.18e90T^{2} \) |
| 23 | \( 1 + 5.06e46T + 4.81e96T^{2} \) |
| 29 | \( 1 - 1.18e52T + 6.76e103T^{2} \) |
| 31 | \( 1 - 7.49e52T + 7.70e105T^{2} \) |
| 37 | \( 1 - 3.49e55T + 2.19e111T^{2} \) |
| 41 | \( 1 - 7.21e56T + 3.21e114T^{2} \) |
| 43 | \( 1 + 1.81e58T + 9.46e115T^{2} \) |
| 47 | \( 1 - 2.91e59T + 5.23e118T^{2} \) |
| 53 | \( 1 + 1.18e61T + 2.65e122T^{2} \) |
| 59 | \( 1 + 4.39e61T + 5.37e125T^{2} \) |
| 61 | \( 1 - 4.52e63T + 5.73e126T^{2} \) |
| 67 | \( 1 + 2.83e64T + 4.48e129T^{2} \) |
| 71 | \( 1 + 3.34e65T + 2.75e131T^{2} \) |
| 73 | \( 1 - 1.11e66T + 1.97e132T^{2} \) |
| 79 | \( 1 + 2.62e67T + 5.38e134T^{2} \) |
| 83 | \( 1 + 2.45e67T + 1.79e136T^{2} \) |
| 89 | \( 1 + 7.46e68T + 2.55e138T^{2} \) |
| 97 | \( 1 + 3.60e70T + 1.15e141T^{2} \) |
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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
−17.29632709408049120438624390575, −14.47150665077030183419276613932, −13.80707754977225033665479531745, −11.87483927420148564700204819479, −9.875233446069549542904010804021, −8.237812742265544899281106525752, −5.66795524891022245294555057796, −4.96037145183897123927359259079, −2.78103252250850026748902801876, −1.06667589081620936244469705442,
1.06667589081620936244469705442, 2.78103252250850026748902801876, 4.96037145183897123927359259079, 5.66795524891022245294555057796, 8.237812742265544899281106525752, 9.875233446069549542904010804021, 11.87483927420148564700204819479, 13.80707754977225033665479531745, 14.47150665077030183419276613932, 17.29632709408049120438624390575
