| L(s) = 1 | − 10.3i·2-s + 12.0·3-s − 44.1·4-s − 27.1·5-s − 125. i·6-s + 631.·7-s − 206. i·8-s − 583.·9-s + 282. i·10-s − 1.42e3i·11-s − 532.·12-s − 1.31e3i·13-s − 6.56e3i·14-s − 327.·15-s − 4.97e3·16-s + 3.33e3·17-s + ⋯ |
| L(s) = 1 | − 1.29i·2-s + 0.447·3-s − 0.689·4-s − 0.217·5-s − 0.581i·6-s + 1.84·7-s − 0.403i·8-s − 0.800·9-s + 0.282i·10-s − 1.06i·11-s − 0.308·12-s − 0.599i·13-s − 2.39i·14-s − 0.0970·15-s − 1.21·16-s + 0.678·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.536i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.843 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.616357 - 2.11859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.616357 - 2.11859i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-1.73e5 + 1.10e5i)T \) |
| good | 2 | \( 1 + 10.3iT - 64T^{2} \) |
| 3 | \( 1 - 12.0T + 729T^{2} \) |
| 5 | \( 1 + 27.1T + 1.56e4T^{2} \) |
| 7 | \( 1 - 631.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.42e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.31e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.33e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 1.22e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 141. iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 798.T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.40e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 7.66e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 6.84e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 4.06e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.48e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.58e5T + 2.21e10T^{2} \) |
| 61 | \( 1 - 1.78e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 1.92e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 1.64e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 6.68e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 3.50e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 6.24e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 7.03e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.11e6iT - 8.32e11T^{2} \) |
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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
−13.40602048254544278632598331444, −11.72198888491125519978992416164, −11.43999852210106497769370452721, −10.27384407455061716695338823799, −8.684897137069266513299297416097, −7.83987169415959283214196657921, −5.47501255585419372775794687249, −3.77590529451141383524395423269, −2.41096016798192428979098829323, −0.921044184957557780408251163895,
2.01651397261087380791126460585, 4.50560612409528512465216074277, 5.65494538453628551524196093032, 7.40505550735955109913183609263, 8.020541231594115899557051958997, 9.118723957007729734234145191227, 11.05729736128249683291228860009, 12.00613150354401551075373867847, 14.00633936708063670418203183272, 14.48730194243056004499344964236
