| L(s) = 1 | + (3.04 + 5.26i)2-s + (−12.8 − 22.1i)3-s + (45.4 − 78.8i)4-s + 442.·5-s + (77.9 − 135. i)6-s + (−380. + 659. i)7-s + 1.33e3·8-s + (765. − 1.32e3i)9-s + (1.34e3 + 2.33e3i)10-s + (−3.05e3 − 5.29e3i)11-s − 2.33e3·12-s + (409. + 7.91e3i)13-s − 4.63e3·14-s + (−5.66e3 − 9.81e3i)15-s + (−1.77e3 − 3.06e3i)16-s + (−1.87e4 + 3.24e4i)17-s + ⋯ |
| L(s) = 1 | + (0.268 + 0.465i)2-s + (−0.274 − 0.474i)3-s + (0.355 − 0.615i)4-s + 1.58·5-s + (0.147 − 0.255i)6-s + (−0.419 + 0.726i)7-s + 0.919·8-s + (0.349 − 0.605i)9-s + (0.425 + 0.736i)10-s + (−0.692 − 1.19i)11-s − 0.389·12-s + (0.0517 + 0.998i)13-s − 0.451·14-s + (−0.433 − 0.751i)15-s + (−0.108 − 0.187i)16-s + (−0.926 + 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.96033 - 0.200282i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96033 - 0.200282i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-409. - 7.91e3i)T \) |
| good | 2 | \( 1 + (-3.04 - 5.26i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (12.8 + 22.1i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 - 442.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (380. - 659. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.05e3 + 5.29e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.87e4 - 3.24e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.69e3 - 2.93e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.49e4 - 2.58e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-2.11e4 - 3.65e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.24e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (7.10e4 + 1.23e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-3.57e4 - 6.19e4i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-6.38e3 + 1.10e4i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 - 4.37e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (8.79e5 - 1.52e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-8.45e5 + 1.46e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.70e6 + 2.96e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-3.97e5 + 6.88e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 3.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.86e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.24e5 - 9.09e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (5.62e5 - 9.75e5i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−18.23166133265438157068368811155, −16.84794466086693935465268337140, −15.42540684811306980332894848759, −13.96087390705276192608088078053, −12.88921131031839686700884913924, −10.79979368025154035855962922978, −9.233244988348446262768065610540, −6.45823747055873561914628892415, −5.77433139074942400397231545469, −1.80131509916613304789953745804,
2.38148296092086042903786576526, 4.89479120310323867881218831809, 7.18313669401892168546047558644, 9.852290193013095526277069878187, 10.74413450552846068201913756051, 12.88118992199962496294326894876, 13.59705091391838080066922066954, 15.80124525480035823020548429754, 17.02288488617192885063595845848, 17.99950541068664605915974076555
