| L(s) = 1 | − 2.73e3·2-s − 1.09e4·3-s − 9.27e6·4-s − 2.86e8i·5-s + 3.00e7·6-s + 2.46e10i·7-s + 7.13e10·8-s − 2.82e11·9-s + 7.85e11i·10-s − 4.90e12i·11-s + 1.01e11·12-s − 1.92e13·13-s − 6.74e13i·14-s + 3.14e12i·15-s − 3.99e13·16-s − 1.36e14i·17-s + ⋯ |
| L(s) = 1 | − 0.668·2-s − 0.0206·3-s − 0.552·4-s − 1.17i·5-s + 0.0137·6-s + 1.77i·7-s + 1.03·8-s − 0.999·9-s + 0.785i·10-s − 1.56i·11-s + 0.0113·12-s − 0.827·13-s − 1.18i·14-s + 0.0242i·15-s − 0.141·16-s − 0.233i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{25}{2})\) |
\(\approx\) |
\(0.5548420788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5548420788\) |
| \(L(13)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-1.62e16 - 1.47e16i)T \) |
| good | 2 | \( 1 + 2.73e3T + 1.67e7T^{2} \) |
| 3 | \( 1 + 1.09e4T + 2.82e11T^{2} \) |
| 5 | \( 1 + 2.86e8iT - 5.96e16T^{2} \) |
| 7 | \( 1 - 2.46e10iT - 1.91e20T^{2} \) |
| 11 | \( 1 + 4.90e12iT - 9.84e24T^{2} \) |
| 13 | \( 1 + 1.92e13T + 5.42e26T^{2} \) |
| 17 | \( 1 + 1.36e14iT - 3.39e29T^{2} \) |
| 19 | \( 1 - 2.47e15iT - 4.89e30T^{2} \) |
| 29 | \( 1 + 6.01e17T + 1.25e35T^{2} \) |
| 31 | \( 1 + 5.44e17T + 6.20e35T^{2} \) |
| 37 | \( 1 - 1.13e19iT - 4.33e37T^{2} \) |
| 41 | \( 1 + 3.48e19T + 5.09e38T^{2} \) |
| 43 | \( 1 - 8.67e18iT - 1.59e39T^{2} \) |
| 47 | \( 1 - 1.32e20T + 1.35e40T^{2} \) |
| 53 | \( 1 + 1.82e20iT - 2.41e41T^{2} \) |
| 59 | \( 1 + 7.47e20T + 3.16e42T^{2} \) |
| 61 | \( 1 - 7.14e20iT - 7.04e42T^{2} \) |
| 67 | \( 1 + 4.33e21iT - 6.69e43T^{2} \) |
| 71 | \( 1 - 2.99e22T + 2.69e44T^{2} \) |
| 73 | \( 1 + 1.89e22T + 5.24e44T^{2} \) |
| 79 | \( 1 + 1.64e22iT - 3.49e45T^{2} \) |
| 83 | \( 1 - 3.52e22iT - 1.14e46T^{2} \) |
| 89 | \( 1 - 1.37e23iT - 6.10e46T^{2} \) |
| 97 | \( 1 - 2.08e23iT - 4.81e47T^{2} \) |
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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
−12.41256870232541937989133729720, −11.36424130419247108234472217920, −9.461810656651725486723737112799, −8.758397863520932630997009211891, −8.154797038919837650680494452489, −5.65983371934544803752309408875, −5.17458493335211665322912777538, −3.23222530983353507799883564913, −1.66993811606580762549508773074, −0.33334107163736145607717842801,
0.50898251573542446443494791007, 2.15489973283334362907287567397, 3.71515669392206649871304860156, 4.88039624421992169308345830987, 7.07537883000380991633449899187, 7.43456200660055703596018902242, 9.237574516499895911784659695328, 10.34511643932101632021620516243, 11.00761470505394700445292219014, 12.96467146346140962817241319662
