| L(s) = 1 | + 1.09e15·2-s + 6.73e23·3-s − 1.61e32·4-s + 1.05e37·5-s + 7.34e38·6-s − 2.03e45·7-s − 3.52e47·8-s − 1.12e51·9-s + 1.15e52·10-s − 7.45e55·11-s − 1.08e56·12-s + 2.59e59·13-s − 2.22e60·14-s + 7.12e60·15-s + 2.57e64·16-s − 7.34e65·17-s − 1.22e66·18-s − 3.57e68·19-s − 1.70e69·20-s − 1.37e69·21-s − 8.13e70·22-s + 7.52e72·23-s − 2.37e71·24-s − 5.04e74·25-s + 2.82e74·26-s − 1.51e75·27-s + 3.28e77·28-s + ⋯ |
| L(s) = 1 | + 0.0856·2-s + 0.0200·3-s − 0.992·4-s + 0.426·5-s + 0.00171·6-s − 1.24·7-s − 0.170·8-s − 0.999·9-s + 0.0364·10-s − 1.43·11-s − 0.0199·12-s + 0.657·13-s − 0.106·14-s + 0.00854·15-s + 0.978·16-s − 1.08·17-s − 0.0855·18-s − 1.37·19-s − 0.422·20-s − 0.0250·21-s − 0.123·22-s + 1.05·23-s − 0.00342·24-s − 0.818·25-s + 0.0562·26-s − 0.0401·27-s + 1.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(108-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+53.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(54)\) |
\(\approx\) |
\(0.3289661175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3289661175\) |
| \(L(\frac{109}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.09e15T + 1.62e32T^{2} \) |
| 3 | \( 1 - 6.73e23T + 1.12e51T^{2} \) |
| 5 | \( 1 - 1.05e37T + 6.16e74T^{2} \) |
| 7 | \( 1 + 2.03e45T + 2.66e90T^{2} \) |
| 11 | \( 1 + 7.45e55T + 2.68e111T^{2} \) |
| 13 | \( 1 - 2.59e59T + 1.55e119T^{2} \) |
| 17 | \( 1 + 7.34e65T + 4.55e131T^{2} \) |
| 19 | \( 1 + 3.57e68T + 6.70e136T^{2} \) |
| 23 | \( 1 - 7.52e72T + 5.06e145T^{2} \) |
| 29 | \( 1 + 2.05e78T + 2.99e156T^{2} \) |
| 31 | \( 1 - 1.28e79T + 3.76e159T^{2} \) |
| 37 | \( 1 + 1.78e83T + 6.27e167T^{2} \) |
| 41 | \( 1 - 1.47e86T + 3.69e172T^{2} \) |
| 43 | \( 1 + 3.56e87T + 6.04e174T^{2} \) |
| 47 | \( 1 + 1.50e89T + 8.21e178T^{2} \) |
| 53 | \( 1 - 1.54e92T + 3.14e184T^{2} \) |
| 59 | \( 1 - 2.89e94T + 3.02e189T^{2} \) |
| 61 | \( 1 + 1.93e95T + 1.07e191T^{2} \) |
| 67 | \( 1 - 5.07e96T + 2.45e195T^{2} \) |
| 71 | \( 1 - 6.83e98T + 1.21e198T^{2} \) |
| 73 | \( 1 - 9.17e99T + 2.37e199T^{2} \) |
| 79 | \( 1 - 4.03e101T + 1.11e203T^{2} \) |
| 83 | \( 1 - 6.14e102T + 2.19e205T^{2} \) |
| 89 | \( 1 + 3.67e104T + 3.84e208T^{2} \) |
| 97 | \( 1 - 1.35e105T + 3.84e212T^{2} \) |
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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
−13.47630361456912755066775941050, −12.95574570342361489013968910328, −10.76784656213115721935679981743, −9.403615782473976886311532878018, −8.343600473452323133117922055882, −6.32025949291226254087366518653, −5.21754026253896477531785081205, −3.63027758635446743160351293587, −2.41207729420044933011581987461, −0.27506538166052387127385127231,
0.27506538166052387127385127231, 2.41207729420044933011581987461, 3.63027758635446743160351293587, 5.21754026253896477531785081205, 6.32025949291226254087366518653, 8.343600473452323133117922055882, 9.403615782473976886311532878018, 10.76784656213115721935679981743, 12.95574570342361489013968910328, 13.47630361456912755066775941050
