| L(s) = 1 | − 4.97i·2-s − 77.4·3-s + 103.·4-s − 105.·5-s + 385. i·6-s − 1.01e3i·7-s − 1.15e3i·8-s + 3.80e3·9-s + 524. i·10-s − 2.57e3i·11-s − 7.99e3·12-s − 4.76e3·13-s − 5.06e3·14-s + 8.16e3·15-s + 7.49e3·16-s − 1.08e4i·17-s + ⋯ |
| L(s) = 1 | − 0.439i·2-s − 1.65·3-s + 0.806·4-s − 0.377·5-s + 0.727i·6-s − 1.12i·7-s − 0.794i·8-s + 1.74·9-s + 0.165i·10-s − 0.582i·11-s − 1.33·12-s − 0.601·13-s − 0.493·14-s + 0.624·15-s + 0.457·16-s − 0.533i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0388638 + 0.0651398i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0388638 + 0.0651398i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (8.42e5 + 1.56e6i)T \) |
| good | 2 | \( 1 + 4.97iT - 128T^{2} \) |
| 3 | \( 1 + 77.4T + 2.18e3T^{2} \) |
| 5 | \( 1 + 105.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.01e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 2.57e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 4.76e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.08e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.55e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.11e5iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 4.64e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.38e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 1.37e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 5.83e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.58e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 7.35e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.56e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.91e6iT - 2.48e12T^{2} \) |
| 67 | \( 1 + 1.17e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.50e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 4.86e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.15e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 1.66e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.48e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 1.03e7T + 8.07e13T^{2} \) |
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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
−12.32920037555580713380161912775, −11.54008371355644399874480350919, −10.83565690662776879240513227551, −9.960159221338161878811493887308, −7.48191606711213584251723513256, −6.65424313831232853483732786654, −5.26991314121457658359957538879, −3.67352115865413790405986823561, −1.30893452248860703818911954797, −0.03516800102462521348424350832,
2.13438759798303295191196998449, 4.72657561325609673080513990821, 5.92789379191299365181444712777, 6.68581681170848575491854960482, 8.112442532807141427051483275994, 10.05006269888022344883109589965, 11.19316036418179361528464017331, 11.99650951068798259458725614253, 12.61070936827899007181558817818, 14.91097861265875317916210754036
