| L(s) = 1 | + 2.26e11·2-s + 2.57e18·3-s − 5.53e23·4-s − 2.99e27·5-s + 5.82e29·6-s − 6.95e31·7-s − 2.62e35·8-s − 4.26e37·9-s − 6.77e38·10-s − 1.05e41·11-s − 1.42e42·12-s + 1.30e44·13-s − 1.57e43·14-s − 7.69e45·15-s + 2.74e47·16-s − 1.86e48·17-s − 9.66e48·18-s + 1.08e50·19-s + 1.65e51·20-s − 1.78e50·21-s − 2.38e52·22-s + 5.47e53·23-s − 6.74e53·24-s − 7.59e54·25-s + 2.94e55·26-s − 2.36e56·27-s + 3.84e55·28-s + ⋯ |
| L(s) = 1 | + 0.291·2-s + 0.366·3-s − 0.915·4-s − 0.735·5-s + 0.106·6-s − 0.0288·7-s − 0.558·8-s − 0.865·9-s − 0.214·10-s − 0.772·11-s − 0.335·12-s + 1.29·13-s − 0.00841·14-s − 0.269·15-s + 0.752·16-s − 0.464·17-s − 0.252·18-s + 0.333·19-s + 0.673·20-s − 0.0105·21-s − 0.224·22-s + 0.891·23-s − 0.204·24-s − 0.458·25-s + 0.378·26-s − 0.683·27-s + 0.0264·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(80-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+79/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(40)\) |
\(\approx\) |
\(1.337194280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.337194280\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.26e11T + 6.04e23T^{2} \) |
| 3 | \( 1 - 2.57e18T + 4.92e37T^{2} \) |
| 5 | \( 1 + 2.99e27T + 1.65e55T^{2} \) |
| 7 | \( 1 + 6.95e31T + 5.79e66T^{2} \) |
| 11 | \( 1 + 1.05e41T + 1.86e82T^{2} \) |
| 13 | \( 1 - 1.30e44T + 1.00e88T^{2} \) |
| 17 | \( 1 + 1.86e48T + 1.60e97T^{2} \) |
| 19 | \( 1 - 1.08e50T + 1.05e101T^{2} \) |
| 23 | \( 1 - 5.47e53T + 3.77e107T^{2} \) |
| 29 | \( 1 - 4.52e57T + 3.38e115T^{2} \) |
| 31 | \( 1 + 5.21e58T + 6.57e117T^{2} \) |
| 37 | \( 1 - 4.17e61T + 7.72e123T^{2} \) |
| 41 | \( 1 - 8.70e63T + 2.56e127T^{2} \) |
| 43 | \( 1 - 6.17e64T + 1.10e129T^{2} \) |
| 47 | \( 1 - 9.80e65T + 1.24e132T^{2} \) |
| 53 | \( 1 + 1.69e68T + 1.65e136T^{2} \) |
| 59 | \( 1 - 1.06e70T + 7.89e139T^{2} \) |
| 61 | \( 1 - 2.41e70T + 1.09e141T^{2} \) |
| 67 | \( 1 + 1.72e72T + 1.81e144T^{2} \) |
| 71 | \( 1 - 8.22e72T + 1.77e146T^{2} \) |
| 73 | \( 1 - 2.63e73T + 1.59e147T^{2} \) |
| 79 | \( 1 + 1.78e75T + 8.17e149T^{2} \) |
| 83 | \( 1 - 3.77e75T + 4.04e151T^{2} \) |
| 89 | \( 1 + 1.69e77T + 1.00e154T^{2} \) |
| 97 | \( 1 - 4.19e78T + 9.01e156T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.75910187455272464037605350458, −14.22147572012564185039407581118, −12.95013077898997502585554581793, −11.14204240724380985306672720402, −9.053325224359333662074186598522, −7.966114034041122198928884801692, −5.71420316279497845240420256637, −4.15203797206811714605804558296, −2.94522297768938828124858349326, −0.63944864586061619364386458998,
0.63944864586061619364386458998, 2.94522297768938828124858349326, 4.15203797206811714605804558296, 5.71420316279497845240420256637, 7.966114034041122198928884801692, 9.053325224359333662074186598522, 11.14204240724380985306672720402, 12.95013077898997502585554581793, 14.22147572012564185039407581118, 15.75910187455272464037605350458
