| L(s) = 1 | + (−4.24 − 5.84i)2-s + (−27.5 − 8.95i)3-s + (23.4 − 72.1i)4-s + (207. − 187. i)5-s + (64.6 + 199. i)6-s + 39.9i·7-s + (−1.40e3 + 454. i)8-s + (−1.09e3 − 791. i)9-s + (−1.97e3 − 416. i)10-s + (−1.62e3 + 1.18e3i)11-s + (−1.29e3 + 1.77e3i)12-s + (−3.87e3 + 5.33e3i)13-s + (233. − 169. i)14-s + (−7.39e3 + 3.30e3i)15-s + (745. + 541. i)16-s + (−6.70e3 + 2.17e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.375 − 0.516i)2-s + (−0.589 − 0.191i)3-s + (0.183 − 0.563i)4-s + (0.742 − 0.670i)5-s + (0.122 + 0.376i)6-s + 0.0439i·7-s + (−0.966 + 0.314i)8-s + (−0.498 − 0.362i)9-s + (−0.624 − 0.131i)10-s + (−0.368 + 0.267i)11-s + (−0.215 + 0.297i)12-s + (−0.489 + 0.673i)13-s + (0.0227 − 0.0165i)14-s + (−0.565 + 0.252i)15-s + (0.0455 + 0.0330i)16-s + (−0.330 + 0.107i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0955489 + 0.653537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0955489 + 0.653537i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-207. + 187. i)T \) |
| good | 2 | \( 1 + (4.24 + 5.84i)T + (-39.5 + 121. i)T^{2} \) |
| 3 | \( 1 + (27.5 + 8.95i)T + (1.76e3 + 1.28e3i)T^{2} \) |
| 7 | \( 1 - 39.9iT - 8.23e5T^{2} \) |
| 11 | \( 1 + (1.62e3 - 1.18e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (3.87e3 - 5.33e3i)T + (-1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (6.70e3 - 2.17e3i)T + (3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (3.79e3 + 1.16e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-2.70e3 - 3.71e3i)T + (-1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-3.76e4 + 1.15e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (3.37e4 + 1.03e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (1.59e4 - 2.19e4i)T + (-2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (6.76e5 + 4.91e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 1.46e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.62e5 - 5.29e4i)T + (4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (1.61e6 + 5.25e5i)T + (9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-6.77e5 - 4.92e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-1.90e6 + 1.38e6i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-3.97e6 + 1.29e6i)T + (4.90e12 - 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-1.50e6 + 4.62e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (2.25e6 + 3.10e6i)T + (-3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-3.62e5 + 1.11e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (8.11e6 - 2.63e6i)T + (2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + (3.56e6 - 2.59e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (2.47e6 + 8.03e5i)T + (6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.36341554459182961832753542014, −13.96762842228071305226222549896, −12.42958768520291505141194613332, −11.35942754022415737791316446075, −9.983223190009634828128968655473, −8.887476619247974786325018662081, −6.45228877672941944983051602633, −5.19737591705972605674117208453, −2.09812546768052958130562186528, −0.38800826133840196170026389678,
2.85374930208261466426650335560, 5.52339899682114117749519786662, 6.88135632594099416563691210275, 8.400630691285088346217505174301, 10.13097079999274618350405440574, 11.34486156263566023910609520376, 12.84614302117907725187083205716, 14.34830270829635846850445989750, 15.72271967067849316184526090122, 16.86932186645329394174935454356
