| L(s) = 1 | + (−7.80 + 7.80i)2-s + 110. i·3-s + 134. i·4-s + (−218. − 218. i)5-s + (−863. − 863. i)6-s − 1.96e3·7-s + (−3.04e3 − 3.04e3i)8-s − 5.69e3·9-s + 3.41e3·10-s + 6.74e3i·11-s − 1.48e4·12-s + (7.24e3 + 7.24e3i)13-s + (1.53e4 − 1.53e4i)14-s + (2.42e4 − 2.42e4i)15-s + 1.31e4·16-s + (4.06e4 + 4.06e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.487 + 0.487i)2-s + 1.36i·3-s + 0.524i·4-s + (−0.350 − 0.350i)5-s + (−0.666 − 0.666i)6-s − 0.817·7-s + (−0.743 − 0.743i)8-s − 0.868·9-s + 0.341·10-s + 0.460i·11-s − 0.716·12-s + (0.253 + 0.253i)13-s + (0.398 − 0.398i)14-s + (0.478 − 0.478i)15-s + 0.200·16-s + (0.486 + 0.486i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.145018 - 0.101915i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.145018 - 0.101915i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (6.55e5 - 1.75e6i)T \) |
| good | 2 | \( 1 + (7.80 - 7.80i)T - 256iT^{2} \) |
| 3 | \( 1 - 110. iT - 6.56e3T^{2} \) |
| 5 | \( 1 + (218. + 218. i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 + 1.96e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 6.74e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (-7.24e3 - 7.24e3i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 + (-4.06e4 - 4.06e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 + (5.74e4 + 5.74e4i)T + 1.69e10iT^{2} \) |
| 23 | \( 1 + (-1.44e3 - 1.44e3i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (-6.13e4 + 6.13e4i)T - 5.00e11iT^{2} \) |
| 31 | \( 1 + (-5.70e5 + 5.70e5i)T - 8.52e11iT^{2} \) |
| 41 | \( 1 - 1.41e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (2.44e6 + 2.44e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + 5.43e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 4.88e5T + 6.22e13T^{2} \) |
| 59 | \( 1 + (5.08e6 + 5.08e6i)T + 1.46e14iT^{2} \) |
| 61 | \( 1 + (-3.15e6 + 3.15e6i)T - 1.91e14iT^{2} \) |
| 67 | \( 1 + 2.26e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 3.52e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 1.97e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (1.40e7 + 1.40e7i)T + 1.51e15iT^{2} \) |
| 83 | \( 1 + 5.02e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-2.78e7 + 2.78e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 + (-6.62e7 - 6.62e7i)T + 7.83e15iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.82127147166374398565728595916, −14.97823979659775522947396299018, −13.09472191262900092613891157580, −11.88775638718175900222609804924, −10.22631970841862907498325471736, −9.311551215121240462252340404361, −8.188205011455193366137660565138, −6.50742800049713209368394786790, −4.50732561466315230189026558344, −3.33957389273116535188413087453,
0.087710064175295094816698158243, 1.37552533933382028224478137959, 2.97898018798399419019619950603, 5.86088594286767643825522150990, 7.00960398842943334104609105896, 8.448718990889000104934728366283, 9.942504776233958824361075526566, 11.23960886394368348900935320251, 12.33911902957790697822085419695, 13.47144502346828302118848860093
