| L(s) = 1 | + (−1.38 + 7.87i)2-s + (16.0 + 13.5i)3-s + (−60.1 − 21.8i)4-s + (358. − 130. i)5-s + (−128. + 108. i)6-s + (−386. − 669. i)7-s + (256 − 443. i)8-s + (−303. − 1.71e3i)9-s + (529. + 3.00e3i)10-s + (1.36e3 − 2.37e3i)11-s + (−672. − 1.16e3i)12-s + (5.54e3 − 4.65e3i)13-s + (5.81e3 − 2.11e3i)14-s + (7.53e3 + 2.74e3i)15-s + (3.13e3 + 2.63e3i)16-s + (−5.76e3 + 3.26e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.344 + 0.288i)3-s + (−0.469 − 0.171i)4-s + (1.28 − 0.466i)5-s + (−0.243 + 0.204i)6-s + (−0.425 − 0.737i)7-s + (0.176 − 0.306i)8-s + (−0.138 − 0.785i)9-s + (0.167 + 0.950i)10-s + (0.310 − 0.537i)11-s + (−0.112 − 0.194i)12-s + (0.700 − 0.587i)13-s + (0.565 − 0.205i)14-s + (0.576 + 0.209i)15-s + (0.191 + 0.160i)16-s + (−0.284 + 1.61i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0523i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.08567 - 0.0546578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08567 - 0.0546578i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 - 7.87i)T \) |
| 19 | \( 1 + (-1.58e4 + 2.53e4i)T \) |
| good | 3 | \( 1 + (-16.0 - 13.5i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (-358. + 130. i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (386. + 669. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.36e3 + 2.37e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-5.54e3 + 4.65e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (5.76e3 - 3.26e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-7.94e4 - 2.88e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (-1.18e4 - 6.72e4i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (8.15e4 + 1.41e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 5.74e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.68e4 - 3.08e4i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-3.31e5 + 1.20e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-8.03e4 - 4.55e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (1.25e6 + 4.56e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (2.78e4 - 1.57e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-2.96e6 - 1.07e6i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-4.58e5 - 2.59e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-3.13e6 + 1.13e6i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (1.51e6 + 1.27e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (2.99e6 + 2.51e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-2.07e6 - 3.58e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.26e6 + 1.05e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (2.22e6 - 1.26e7i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−14.79339198036341822999934420870, −13.62114704682794987709969417836, −12.93755045225106827709362470482, −10.69976936533878770118902788759, −9.464272080783837663887076085387, −8.644639725717804925712038147203, −6.69662462131730538967186230357, −5.57587865339931351437118060709, −3.61455382428844769256560156744, −1.02905041496604410666620367609,
1.76464983617500877379023533771, 2.86099147380785973820902605052, 5.27620754469015590354271430307, 6.90285691535366681206873316264, 8.851934459339454983195811130318, 9.747147922847654924395789284371, 11.04151907248951126732112215323, 12.45595336676900356661026923940, 13.67487252236869649721483773781, 14.24383956273823273791496629955
