| L(s) = 1 | + (2.63e6 − 3.26e6i)2-s + 5.11e9i·3-s + (−3.68e12 − 1.72e13i)4-s − 1.83e15·5-s + (1.66e16 + 1.34e16i)6-s − 1.24e18i·7-s + (−6.58e19 − 3.33e19i)8-s + 9.58e20·9-s + (−4.84e21 + 5.98e21i)10-s − 1.41e23i·11-s + (8.79e22 − 1.88e22i)12-s − 3.73e24·13-s + (−4.06e24 − 3.28e24i)14-s − 9.38e24i·15-s + (−2.82e26 + 1.26e26i)16-s + 9.62e26·17-s + ⋯ |
| L(s) = 1 | + (0.628 − 0.777i)2-s + 0.162i·3-s + (−0.209 − 0.977i)4-s − 0.770·5-s + (0.126 + 0.102i)6-s − 0.318i·7-s + (−0.892 − 0.451i)8-s + 0.973·9-s + (−0.484 + 0.598i)10-s − 1.73i·11-s + (0.159 − 0.0341i)12-s − 1.16·13-s + (−0.247 − 0.200i)14-s − 0.125i·15-s + (−0.912 + 0.409i)16-s + 0.819·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(45-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+22) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{45}{2})\) |
\(\approx\) |
\(0.3150298706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3150298706\) |
| \(L(23)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.63e6 + 3.26e6i)T \) |
| good | 3 | \( 1 - 5.11e9iT - 9.84e20T^{2} \) |
| 5 | \( 1 + 1.83e15T + 5.68e30T^{2} \) |
| 7 | \( 1 + 1.24e18iT - 1.52e37T^{2} \) |
| 11 | \( 1 + 1.41e23iT - 6.62e45T^{2} \) |
| 13 | \( 1 + 3.73e24T + 1.03e49T^{2} \) |
| 17 | \( 1 - 9.62e26T + 1.37e54T^{2} \) |
| 19 | \( 1 - 7.45e27iT - 1.84e56T^{2} \) |
| 23 | \( 1 - 7.93e29iT - 8.24e59T^{2} \) |
| 29 | \( 1 + 2.33e32T + 2.21e64T^{2} \) |
| 31 | \( 1 - 4.48e31iT - 4.16e65T^{2} \) |
| 37 | \( 1 - 1.89e34T + 1.00e69T^{2} \) |
| 41 | \( 1 + 1.40e35T + 9.17e70T^{2} \) |
| 43 | \( 1 - 4.14e35iT - 7.45e71T^{2} \) |
| 47 | \( 1 - 6.22e36iT - 3.73e73T^{2} \) |
| 53 | \( 1 + 5.66e37T + 7.38e75T^{2} \) |
| 59 | \( 1 - 1.58e39iT - 8.26e77T^{2} \) |
| 61 | \( 1 + 1.71e39T + 3.58e78T^{2} \) |
| 67 | \( 1 + 1.52e39iT - 2.22e80T^{2} \) |
| 71 | \( 1 + 5.26e40iT - 2.85e81T^{2} \) |
| 73 | \( 1 - 1.04e40T + 9.68e81T^{2} \) |
| 79 | \( 1 + 9.56e41iT - 3.13e83T^{2} \) |
| 83 | \( 1 - 2.60e42iT - 2.75e84T^{2} \) |
| 89 | \( 1 + 8.08e42T + 5.93e85T^{2} \) |
| 97 | \( 1 - 1.49e43T + 2.61e87T^{2} \) |
| 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
−13.59963757794294174449701679085, −12.17918113185564577444764938711, −10.96634742737830741467507870658, −9.607627013765662916220607478734, −7.60762907406117819143479054566, −5.67268923073270960556232288887, −4.15433897659077121845474108859, −3.19767319663320608582352157741, −1.34338978916771176861350199351, −0.06861134077647163687549871883,
2.21412528086000797226096915695, 4.00915447488166447720199212903, 5.02051359277398581135902019997, 6.99118380974996308471205876936, 7.71071505849247338464671616416, 9.659059536732959385500987105281, 12.03395336168938900183308723869, 12.79686170870166627770982776401, 14.76435124379400235857499451726, 15.53386248257781804060050583490
