| L(s) = 1 | + 1.24e10·2-s + 1.66e16·3-s − 4.36e20·4-s + 1.87e23·5-s + 2.07e26·6-s − 1.62e29·7-s − 1.27e31·8-s + 2.78e32·9-s + 2.33e33·10-s − 2.28e35·11-s − 7.27e36·12-s − 3.32e38·13-s − 2.01e39·14-s + 3.12e39·15-s + 9.90e40·16-s + 1.24e42·17-s + 3.45e42·18-s + 4.82e42·19-s − 8.18e43·20-s − 2.70e45·21-s − 2.84e45·22-s + 4.28e46·23-s − 2.12e47·24-s − 1.65e48·25-s − 4.13e48·26-s + 4.63e48·27-s + 7.07e49·28-s + ⋯ |
| L(s) = 1 | + 0.511·2-s + 0.577·3-s − 0.738·4-s + 0.144·5-s + 0.295·6-s − 1.13·7-s − 0.888·8-s + 0.333·9-s + 0.0737·10-s − 0.270·11-s − 0.426·12-s − 1.23·13-s − 0.579·14-s + 0.0832·15-s + 0.284·16-s + 0.440·17-s + 0.170·18-s + 0.0368·19-s − 0.106·20-s − 0.654·21-s − 0.138·22-s + 0.449·23-s − 0.513·24-s − 0.979·25-s − 0.630·26-s + 0.192·27-s + 0.837·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(70-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+69/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(35)\) |
\(\approx\) |
\(1.751993911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751993911\) |
| \(L(\frac{71}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.66e16T \) |
| good | 2 | \( 1 - 1.24e10T + 5.90e20T^{2} \) |
| 5 | \( 1 - 1.87e23T + 1.69e48T^{2} \) |
| 7 | \( 1 + 1.62e29T + 2.05e58T^{2} \) |
| 11 | \( 1 + 2.28e35T + 7.17e71T^{2} \) |
| 13 | \( 1 + 3.32e38T + 7.27e76T^{2} \) |
| 17 | \( 1 - 1.24e42T + 7.96e84T^{2} \) |
| 19 | \( 1 - 4.82e42T + 1.71e88T^{2} \) |
| 23 | \( 1 - 4.28e46T + 9.10e93T^{2} \) |
| 29 | \( 1 + 2.67e49T + 8.04e100T^{2} \) |
| 31 | \( 1 - 2.16e51T + 8.01e102T^{2} \) |
| 37 | \( 1 - 1.09e54T + 1.60e108T^{2} \) |
| 41 | \( 1 - 4.37e55T + 1.91e111T^{2} \) |
| 43 | \( 1 + 1.86e56T + 5.12e112T^{2} \) |
| 47 | \( 1 - 9.54e57T + 2.37e115T^{2} \) |
| 53 | \( 1 + 3.08e59T + 9.44e118T^{2} \) |
| 59 | \( 1 + 4.56e59T + 1.54e122T^{2} \) |
| 61 | \( 1 + 6.05e61T + 1.54e123T^{2} \) |
| 67 | \( 1 - 4.99e62T + 9.98e125T^{2} \) |
| 71 | \( 1 - 1.22e64T + 5.45e127T^{2} \) |
| 73 | \( 1 + 9.97e63T + 3.70e128T^{2} \) |
| 79 | \( 1 + 1.09e65T + 8.63e130T^{2} \) |
| 83 | \( 1 - 1.11e66T + 2.60e132T^{2} \) |
| 89 | \( 1 - 2.75e67T + 3.22e134T^{2} \) |
| 97 | \( 1 + 2.52e67T + 1.22e137T^{2} \) |
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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
−13.20204554378101564606869573746, −12.27357144983333272001464642346, −9.982251693553581393593352721404, −9.229190236883394997758235306177, −7.66549194850513822293746593027, −6.08365504372772326883736904307, −4.73401290924350088938580149437, −3.48087728370306431655981435195, −2.51068315269915549079655997119, −0.56981536586595189697480263143,
0.56981536586595189697480263143, 2.51068315269915549079655997119, 3.48087728370306431655981435195, 4.73401290924350088938580149437, 6.08365504372772326883736904307, 7.66549194850513822293746593027, 9.229190236883394997758235306177, 9.982251693553581393593352721404, 12.27357144983333272001464642346, 13.20204554378101564606869573746
