| L(s) = 1 | − 2.79e10·2-s + 1.66e16·3-s + 1.91e20·4-s − 1.83e24·5-s − 4.66e26·6-s − 3.84e28·7-s + 1.11e31·8-s + 2.78e32·9-s + 5.12e34·10-s + 1.40e35·11-s + 3.19e36·12-s − 6.79e37·13-s + 1.07e39·14-s − 3.05e40·15-s − 4.24e41·16-s + 4.71e41·17-s − 7.77e42·18-s − 1.78e44·19-s − 3.51e44·20-s − 6.40e44·21-s − 3.92e45·22-s + 2.21e46·23-s + 1.85e47·24-s + 1.66e48·25-s + 1.90e48·26-s + 4.63e48·27-s − 7.36e48·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.577·3-s + 0.324·4-s − 1.40·5-s − 0.664·6-s − 0.268·7-s + 0.777·8-s + 0.333·9-s + 1.62·10-s + 0.165·11-s + 0.187·12-s − 0.251·13-s + 0.308·14-s − 0.812·15-s − 1.21·16-s + 0.167·17-s − 0.383·18-s − 1.36·19-s − 0.457·20-s − 0.154·21-s − 0.190·22-s + 0.231·23-s + 0.448·24-s + 0.982·25-s + 0.289·26-s + 0.192·27-s − 0.0871·28-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
| \(L(35)\) |
\(\approx\) |
\(0.4051152945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4051152945\) |
| \(L(\frac{71}{2})\) |
|
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 | \( 1 + 2.79e10T + 5.90e20T^{2} \) |
| 5 | \( 1 + 1.83e24T + 1.69e48T^{2} \) |
| 7 | \( 1 + 3.84e28T + 2.05e58T^{2} \) |
| 11 | \( 1 - 1.40e35T + 7.17e71T^{2} \) |
| 13 | \( 1 + 6.79e37T + 7.27e76T^{2} \) |
| 17 | \( 1 - 4.71e41T + 7.96e84T^{2} \) |
| 19 | \( 1 + 1.78e44T + 1.71e88T^{2} \) |
| 23 | \( 1 - 2.21e46T + 9.10e93T^{2} \) |
| 29 | \( 1 + 4.79e50T + 8.04e100T^{2} \) |
| 31 | \( 1 - 2.88e51T + 8.01e102T^{2} \) |
| 37 | \( 1 + 9.12e53T + 1.60e108T^{2} \) |
| 41 | \( 1 + 7.69e55T + 1.91e111T^{2} \) |
| 43 | \( 1 - 3.90e55T + 5.12e112T^{2} \) |
| 47 | \( 1 + 2.29e57T + 2.37e115T^{2} \) |
| 53 | \( 1 + 4.81e59T + 9.44e118T^{2} \) |
| 59 | \( 1 - 1.21e61T + 1.54e122T^{2} \) |
| 61 | \( 1 - 7.30e61T + 1.54e123T^{2} \) |
| 67 | \( 1 + 3.48e62T + 9.98e125T^{2} \) |
| 71 | \( 1 + 1.11e64T + 5.45e127T^{2} \) |
| 73 | \( 1 + 1.42e64T + 3.70e128T^{2} \) |
| 79 | \( 1 - 1.41e65T + 8.63e130T^{2} \) |
| 83 | \( 1 + 2.98e66T + 2.60e132T^{2} \) |
| 89 | \( 1 - 2.83e67T + 3.22e134T^{2} \) |
| 97 | \( 1 - 4.20e68T + 1.22e137T^{2} \) |
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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
−13.01672668871918946816809496880, −11.46287024998856903410232600835, −10.11815053329678930067745541092, −8.787682772890809300315129111079, −7.987468604282412420187038734879, −6.94370111043086827849156458544, −4.54099060560191992326640139429, −3.45334893702170339260922784269, −1.80577698387341244129524003488, −0.36076670471800663170813780980,
0.36076670471800663170813780980, 1.80577698387341244129524003488, 3.45334893702170339260922784269, 4.54099060560191992326640139429, 6.94370111043086827849156458544, 7.987468604282412420187038734879, 8.787682772890809300315129111079, 10.11815053329678930067745541092, 11.46287024998856903410232600835, 13.01672668871918946816809496880
