| L(s) = 1 | + (−0.670 − 0.562i)3-s + (−12.4 + 4.54i)5-s + (41.6 + 72.1i)7-s + (−42.0 − 238. i)9-s + (−226. + 392. i)11-s + (−370. + 311. i)13-s + (10.9 + 3.98i)15-s + (−138. + 784. i)17-s + (−2.60 + 1.57e3i)19-s + (12.6 − 71.8i)21-s + (−3.26e3 − 1.18e3i)23-s + (−2.25e3 + 1.89e3i)25-s + (−212. + 367. i)27-s + (1.08e3 + 6.14e3i)29-s + (816. + 1.41e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.0430 − 0.0361i)3-s + (−0.223 + 0.0813i)5-s + (0.321 + 0.556i)7-s + (−0.173 − 0.981i)9-s + (−0.564 + 0.978i)11-s + (−0.608 + 0.510i)13-s + (0.0125 + 0.00456i)15-s + (−0.116 + 0.658i)17-s + (−0.00165 + 0.999i)19-s + (0.00627 − 0.0355i)21-s + (−1.28 − 0.468i)23-s + (−0.722 + 0.606i)25-s + (−0.0560 + 0.0971i)27-s + (0.239 + 1.35i)29-s + (0.152 + 0.264i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.354282 + 0.742763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.354282 + 0.742763i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.60 - 1.57e3i)T \) |
| good | 3 | \( 1 + (0.670 + 0.562i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (12.4 - 4.54i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-41.6 - 72.1i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (226. - 392. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (370. - 311. i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (138. - 784. i)T + (-1.33e6 - 4.85e5i)T^{2} \) |
| 23 | \( 1 + (3.26e3 + 1.18e3i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-1.08e3 - 6.14e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-816. - 1.41e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 4.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.17e4 + 9.85e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-1.98e4 + 7.21e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-961. - 5.45e3i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-1.93e4 - 7.03e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-6.80e3 + 3.86e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (2.48e4 + 9.05e3i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-725. - 4.11e3i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-3.52e4 + 1.28e4i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 + (-2.43e4 - 2.03e4i)T + (3.59e8 + 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-2.64e4 - 2.22e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (2.05e3 + 3.55e3i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (8.08e4 - 6.78e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (1.65e4 - 9.38e4i)T + (-8.06e9 - 2.93e9i)T^{2} \) |
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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
−14.12540269511019726459071256689, −12.36778538223960133216422417659, −12.12704337479367269459731167145, −10.54319569814747265512281352682, −9.421984713530027319600022953447, −8.181185151835493315111541653898, −6.87726048482456800270567104028, −5.46771960181158287839606151774, −3.88947767740541887118170355625, −1.99969873333225412032052791114,
0.34052154620096585340046808917, 2.57437535208153957560433619775, 4.43884667880636285562832551727, 5.71900416184213864859151254667, 7.51009701704537745231434686195, 8.275100032597687167076760213153, 9.922796215147623426156173061270, 10.94269775136352749107250025591, 11.87862995475906524642073299362, 13.47929246322291143204664839612
