| L(s) = 1 | + (2 − 3.46i)2-s + (−7.99 − 13.8i)4-s + (11.2 − 19.4i)5-s + (−96.4 + 86.5i)7-s − 63.9·8-s + (−44.9 − 77.9i)10-s + (170. + 294. i)11-s − 728.·13-s + (106. + 507. i)14-s + (−128 + 221. i)16-s + (404. + 701. i)17-s + (−513. + 888. i)19-s − 359.·20-s + 1.36e3·22-s + (711. − 1.23e3i)23-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.201 − 0.348i)5-s + (−0.744 + 0.667i)7-s − 0.353·8-s + (−0.142 − 0.246i)10-s + (0.424 + 0.734i)11-s − 1.19·13-s + (0.145 + 0.691i)14-s + (−0.125 + 0.216i)16-s + (0.339 + 0.588i)17-s + (−0.326 + 0.564i)19-s − 0.201·20-s + 0.599·22-s + (0.280 − 0.485i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.060121984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060121984\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (96.4 - 86.5i)T \) |
| good | 5 | \( 1 + (-11.2 + 19.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-170. - 294. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 728.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-404. - 701. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (513. - 888. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-711. + 1.23e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 5.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-3.51e3 - 6.09e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (6.39e3 - 1.10e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-4.65e3 + 8.06e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.78e4 - 3.08e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.51e4 + 2.63e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.60e4 - 2.78e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.06e4 + 1.84e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (2.06e4 + 3.57e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.75e4 + 3.03e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 8.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-3.89e4 + 6.75e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.61e5T + 8.58e9T^{2} \) |
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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
−12.38397056920781184405751426184, −12.10744426207410414675035052999, −10.48792031285069340291983896714, −9.652441778762173953483579197447, −8.740659253089825151546420509926, −7.06833211016014294132596465115, −5.76361348712872961936788925217, −4.59976368735826400567093840977, −3.07636583610418081012046819235, −1.68054151491906424939103888895,
0.32654658014322787206962656965, 2.79605513249982321686827325637, 4.12457067199298598192794404726, 5.59202045253676122459458449674, 6.76495015722998805396506992118, 7.54755158398408503166965863555, 9.069879192122574898495856516542, 10.01083363831106342268891736571, 11.25742572585084114353443442387, 12.44960581179986635913304281894
