| L(s) = 1 | − 5.81e10·2-s − 8.58e16·3-s + 1.01e21·4-s + 8.21e23·5-s + 4.98e27·6-s − 4.94e29·7-s + 7.82e31·8-s − 1.42e32·9-s − 4.77e34·10-s − 1.31e37·11-s − 8.71e37·12-s − 1.82e39·13-s + 2.87e40·14-s − 7.05e40·15-s − 6.94e42·16-s − 9.13e43·17-s + 8.25e42·18-s − 2.40e45·19-s + 8.33e44·20-s + 4.24e46·21-s + 7.61e47·22-s − 1.18e48·23-s − 6.71e48·24-s − 4.16e49·25-s + 1.06e50·26-s + 6.56e50·27-s − 5.01e50·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 0.990·3-s + 0.429·4-s + 0.126·5-s + 1.18·6-s − 0.493·7-s + 0.681·8-s − 0.0189·9-s − 0.150·10-s − 1.40·11-s − 0.425·12-s − 0.520·13-s + 0.589·14-s − 0.125·15-s − 1.24·16-s − 1.90·17-s + 0.0226·18-s − 0.964·19-s + 0.0542·20-s + 0.488·21-s + 1.68·22-s − 0.539·23-s − 0.675·24-s − 0.984·25-s + 0.622·26-s + 1.00·27-s − 0.212·28-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
| \(L(36)\) |
\(\approx\) |
\(7.150662917\times10^{-6}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.150662917\times10^{-6}\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 5.81e10T + 2.36e21T^{2} \) |
| 3 | \( 1 + 8.58e16T + 7.50e33T^{2} \) |
| 5 | \( 1 - 8.21e23T + 4.23e49T^{2} \) |
| 7 | \( 1 + 4.94e29T + 1.00e60T^{2} \) |
| 11 | \( 1 + 1.31e37T + 8.68e73T^{2} \) |
| 13 | \( 1 + 1.82e39T + 1.23e79T^{2} \) |
| 17 | \( 1 + 9.13e43T + 2.30e87T^{2} \) |
| 19 | \( 1 + 2.40e45T + 6.18e90T^{2} \) |
| 23 | \( 1 + 1.18e48T + 4.81e96T^{2} \) |
| 29 | \( 1 - 4.89e51T + 6.76e103T^{2} \) |
| 31 | \( 1 + 3.46e52T + 7.70e105T^{2} \) |
| 37 | \( 1 - 4.69e55T + 2.19e111T^{2} \) |
| 41 | \( 1 + 2.90e57T + 3.21e114T^{2} \) |
| 43 | \( 1 + 7.18e57T + 9.46e115T^{2} \) |
| 47 | \( 1 + 1.61e59T + 5.23e118T^{2} \) |
| 53 | \( 1 + 2.34e61T + 2.65e122T^{2} \) |
| 59 | \( 1 - 2.94e62T + 5.37e125T^{2} \) |
| 61 | \( 1 - 2.30e63T + 5.73e126T^{2} \) |
| 67 | \( 1 + 1.82e64T + 4.48e129T^{2} \) |
| 71 | \( 1 - 1.03e66T + 2.75e131T^{2} \) |
| 73 | \( 1 + 9.40e64T + 1.97e132T^{2} \) |
| 79 | \( 1 + 3.07e67T + 5.38e134T^{2} \) |
| 83 | \( 1 - 1.15e68T + 1.79e136T^{2} \) |
| 89 | \( 1 + 1.27e69T + 2.55e138T^{2} \) |
| 97 | \( 1 - 2.40e70T + 1.15e141T^{2} \) |
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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
−17.24686278890259140812859640143, −15.87880331422277564722801333734, −13.14383135487989516343864605052, −11.13392762698162904441864648134, −9.980976733636039742866100511736, −8.297474605567912177455966516506, −6.54386990840114140755872850028, −4.82425745080437901523267684343, −2.18585290000314499876701611719, −0.00194874992571728991050332610,
0.00194874992571728991050332610, 2.18585290000314499876701611719, 4.82425745080437901523267684343, 6.54386990840114140755872850028, 8.297474605567912177455966516506, 9.980976733636039742866100511736, 11.13392762698162904441864648134, 13.14383135487989516343864605052, 15.87880331422277564722801333734, 17.24686278890259140812859640143
