| L(s) = 1 | − 1.34e10·2-s + 7.87e15·3-s + 3.28e19·4-s − 3.01e23·5-s − 1.05e26·6-s − 4.20e26·7-s + 1.54e30·8-s − 3.06e31·9-s + 4.04e33·10-s + 1.62e34·11-s + 2.59e35·12-s − 3.08e37·13-s + 5.65e36·14-s − 2.37e39·15-s − 2.55e40·16-s − 1.59e41·17-s + 4.11e41·18-s + 1.03e43·19-s − 9.91e42·20-s − 3.31e42·21-s − 2.17e44·22-s − 3.59e45·23-s + 1.21e46·24-s + 2.30e46·25-s + 4.14e47·26-s − 9.71e47·27-s − 1.38e46·28-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.818·3-s + 0.222·4-s − 1.15·5-s − 0.904·6-s − 0.0205·7-s + 0.859·8-s − 0.330·9-s + 1.28·10-s + 0.210·11-s + 0.182·12-s − 1.48·13-s + 0.0227·14-s − 0.947·15-s − 1.17·16-s − 0.960·17-s + 0.365·18-s + 1.50·19-s − 0.258·20-s − 0.0168·21-s − 0.232·22-s − 0.865·23-s + 0.703·24-s + 0.340·25-s + 1.64·26-s − 1.08·27-s − 0.00458·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(68-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+67/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(34)\) |
\(\approx\) |
\(0.7305257436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7305257436\) |
| \(L(\frac{69}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.34e10T + 1.47e20T^{2} \) |
| 3 | \( 1 - 7.87e15T + 9.27e31T^{2} \) |
| 5 | \( 1 + 3.01e23T + 6.77e46T^{2} \) |
| 7 | \( 1 + 4.20e26T + 4.18e56T^{2} \) |
| 11 | \( 1 - 1.62e34T + 5.93e69T^{2} \) |
| 13 | \( 1 + 3.08e37T + 4.30e74T^{2} \) |
| 17 | \( 1 + 1.59e41T + 2.75e82T^{2} \) |
| 19 | \( 1 - 1.03e43T + 4.74e85T^{2} \) |
| 23 | \( 1 + 3.59e45T + 1.72e91T^{2} \) |
| 29 | \( 1 - 1.54e49T + 9.56e97T^{2} \) |
| 31 | \( 1 - 1.52e50T + 8.34e99T^{2} \) |
| 37 | \( 1 - 1.16e52T + 1.17e105T^{2} \) |
| 41 | \( 1 - 1.50e54T + 1.13e108T^{2} \) |
| 43 | \( 1 - 1.43e54T + 2.76e109T^{2} \) |
| 47 | \( 1 - 9.41e55T + 1.07e112T^{2} \) |
| 53 | \( 1 - 4.49e57T + 3.36e115T^{2} \) |
| 59 | \( 1 - 2.81e59T + 4.43e118T^{2} \) |
| 61 | \( 1 + 4.42e59T + 4.14e119T^{2} \) |
| 67 | \( 1 - 9.71e60T + 2.22e122T^{2} \) |
| 71 | \( 1 + 3.66e61T + 1.08e124T^{2} \) |
| 73 | \( 1 + 1.52e61T + 6.96e124T^{2} \) |
| 79 | \( 1 + 2.59e63T + 1.38e127T^{2} \) |
| 83 | \( 1 + 2.25e64T + 3.78e128T^{2} \) |
| 89 | \( 1 + 2.92e64T + 4.06e130T^{2} \) |
| 97 | \( 1 + 8.79e65T + 1.29e133T^{2} \) |
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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
−17.59882734512744097320727731167, −15.81448367747186753295575471035, −14.07728569605045425316424108796, −11.73931573546291553117159630696, −9.755535798977241030557905771745, −8.391833007584314591458868850339, −7.43183866614746778765322778546, −4.38977130502919495322892275986, −2.60914844321186924080308460842, −0.60391611597935734051522552277,
0.60391611597935734051522552277, 2.60914844321186924080308460842, 4.38977130502919495322892275986, 7.43183866614746778765322778546, 8.391833007584314591458868850339, 9.755535798977241030557905771745, 11.73931573546291553117159630696, 14.07728569605045425316424108796, 15.81448367747186753295575471035, 17.59882734512744097320727731167
