| L(s) = 1 | + (7.31 + 5.31i)2-s + (−0.760 − 2.33i)3-s + (15.4 + 47.3i)4-s + (−10.6 + 54.8i)5-s + (6.87 − 21.1i)6-s + 88.6·7-s + (−49.8 + 153. i)8-s + (191. − 139. i)9-s + (−370. + 344. i)10-s + (−521. − 379. i)11-s + (99.1 − 72.0i)12-s + (47.0 − 34.1i)13-s + (648. + 471. i)14-s + (136. − 16.6i)15-s + (109. − 79.4i)16-s + (331. − 1.02e3i)17-s + ⋯ |
| L(s) = 1 | + (1.29 + 0.939i)2-s + (−0.0487 − 0.150i)3-s + (0.481 + 1.48i)4-s + (−0.191 + 0.981i)5-s + (0.0779 − 0.240i)6-s + 0.683·7-s + (−0.275 + 0.847i)8-s + (0.788 − 0.573i)9-s + (−1.17 + 1.08i)10-s + (−1.29 − 0.944i)11-s + (0.198 − 0.144i)12-s + (0.0771 − 0.0560i)13-s + (0.884 + 0.642i)14-s + (0.156 − 0.0191i)15-s + (0.106 − 0.0775i)16-s + (0.278 − 0.856i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.08950 + 1.71420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08950 + 1.71420i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (10.6 - 54.8i)T \) |
| good | 2 | \( 1 + (-7.31 - 5.31i)T + (9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (0.760 + 2.33i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 - 88.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + (521. + 379. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-47.0 + 34.1i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-331. + 1.02e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (362. - 1.11e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-541. - 393. i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-656. - 2.01e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.84e3 - 5.69e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (9.03e3 - 6.56e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.44e4 + 1.04e4i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.82e3 + 8.69e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.44e3 - 4.43e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.22e4 - 8.88e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.76e4 - 2.73e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.82e4 + 5.61e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.06e4 + 3.28e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.50e4 + 1.81e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.74e4 - 8.44e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-5.16e3 + 1.58e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-8.33e4 - 6.05e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-2.60e4 - 8.02e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−16.23011625153665004992905121091, −15.37866074829472514755388297313, −14.36845175306810009221524510264, −13.40502874904748946400266408526, −12.03721725316891470794472229516, −10.50651071753815344793944403145, −7.893229364705455603923929298362, −6.75880890119395468698521716532, −5.25101959333006923351227308037, −3.38686946219453087001953293481,
1.91496062610742696968045619831, 4.35940380283678731536510856907, 5.20525225224142046655470259341, 7.946702849427919288617923976635, 10.11016554333512266158385397745, 11.32451220076186738062260784491, 12.75240024752148694896080985554, 13.17940287257147853789450084136, 14.82577050274405763796591054337, 15.87931402457231279523567637302
