| L(s) = 1 | + 5.65·2-s − 15.5·3-s + 32.0·4-s − 223. i·5-s − 88.1·6-s − 652. i·7-s + 181.·8-s + 243·9-s − 1.26e3i·10-s + 524. i·11-s − 498.·12-s − 1.11e3·13-s − 3.69e3i·14-s + 3.47e3i·15-s + 1.02e3·16-s − 2.16e3i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.500·4-s − 1.78i·5-s − 0.408·6-s − 1.90i·7-s + 0.353·8-s + 0.333·9-s − 1.26i·10-s + 0.394i·11-s − 0.288·12-s − 0.508·13-s − 1.34i·14-s + 1.03i·15-s + 0.250·16-s − 0.440i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.811730713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.811730713\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65T \) |
| 3 | \( 1 + 15.5T \) |
| 23 | \( 1 + (1.20e4 - 1.58e3i)T \) |
| good | 5 | \( 1 + 223. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 652. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 524. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.11e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 2.16e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.03e4iT - 4.70e7T^{2} \) |
| 29 | \( 1 - 3.45e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.67e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 6.31e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.36e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 9.90e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 9.91e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 9.89e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.55e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 3.24e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 5.99e3iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 6.26e4T + 1.28e11T^{2} \) |
| 73 | \( 1 - 3.53e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.16e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.61e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.90e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.16e6iT - 8.32e11T^{2} \) |
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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
−12.06165348124946492974900433697, −10.52250560205355612917814822411, −9.786751603696495359957208285423, −8.126969176607803620561382806686, −7.18104630080250402699071758726, −5.73791231945474044623036017234, −4.57570111317257763977776352097, −4.04919763892604918311770947132, −1.48639870989588244313278739174, −0.46731971730472939789565238615,
2.34085426623931906751625336565, 3.04592758517054443738594276353, 4.90047846760245512615040827693, 6.17478793814369040170450168011, 6.59629856074604838911055386920, 8.139805849469718097236664808924, 9.720815030875773096811323677831, 10.81374328024078808286196283902, 11.65650348166681662583638303537, 12.24291843049083887253368531915
