| L(s) = 1 | + (5.05 + 8.75i)2-s + (35.2 + 60.9i)3-s + (12.8 − 22.2i)4-s − 163.·5-s + (−356. + 616. i)6-s + (−123. + 213. i)7-s + 1.55e3·8-s + (−1.38e3 + 2.39e3i)9-s + (−829. − 1.43e3i)10-s + (−1.01e3 − 1.75e3i)11-s + 1.80e3·12-s + (7.17e3 − 3.35e3i)13-s − 2.49e3·14-s + (−5.77e3 − 9.99e3i)15-s + (6.21e3 + 1.07e4i)16-s + (1.74e4 − 3.02e4i)17-s + ⋯ |
| L(s) = 1 | + (0.446 + 0.774i)2-s + (0.752 + 1.30i)3-s + (0.100 − 0.173i)4-s − 0.586·5-s + (−0.672 + 1.16i)6-s + (−0.136 + 0.235i)7-s + 1.07·8-s + (−0.633 + 1.09i)9-s + (−0.262 − 0.454i)10-s + (−0.230 − 0.398i)11-s + 0.302·12-s + (0.905 − 0.423i)13-s − 0.243·14-s + (−0.441 − 0.764i)15-s + (0.379 + 0.657i)16-s + (0.863 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.41708 + 1.70238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41708 + 1.70238i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-7.17e3 + 3.35e3i)T \) |
| good | 2 | \( 1 + (-5.05 - 8.75i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-35.2 - 60.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 163.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (123. - 213. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.01e3 + 1.75e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.74e4 + 3.02e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.07e4 - 1.86e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.51e4 - 6.08e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.03e5 + 1.78e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 3.04e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-6.13e4 - 1.06e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (1.17e5 + 2.03e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.56e4 - 7.91e4i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 9.67e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.87e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + (3.82e4 - 6.62e4i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.96e5 + 5.13e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-9.15e5 - 1.58e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.41e6 + 2.45e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 1.74e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.23e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.95e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.54e6 - 4.40e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (7.53e6 - 1.30e7i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
| 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
−18.88082337363426179486743876919, −16.44299601134327849059698787080, −15.71351870571114994353155392453, −14.87840111964350827899828698580, −13.63944559615453532333879442114, −11.13207005895577564586018885969, −9.602052074659595272574537113657, −7.83761993297950637049855201632, −5.45076638986298220502432034079, −3.65368900467429554289915085310,
1.73011837252822498245577459623, 3.59607283448706292910255703133, 7.07336017669794773128925861744, 8.332288172435738107106387343369, 10.97422961153423946574092356201, 12.53026337896966934259529855100, 13.14336529871116976012551457263, 14.63324112819501566213269137165, 16.60991029552748832556878498012, 18.42347940652428924436752056817
