| L(s) = 1 | + (13.7 + 8.86i)2-s + (−11.8 − 82.6i)3-s + (58.4 + 128. i)4-s + (−324. − 95.3i)5-s + (568. − 1.24e3i)6-s + (482. − 556. i)7-s + (−29.6 + 206. i)8-s + (−4.58e3 + 1.34e3i)9-s + (−3.63e3 − 4.19e3i)10-s + (2.70e3 − 1.73e3i)11-s + (9.88e3 − 6.35e3i)12-s + (5.94e3 + 6.85e3i)13-s + (1.15e4 − 3.40e3i)14-s + (−4.02e3 + 2.79e4i)15-s + (9.55e3 − 1.10e4i)16-s + (8.99e3 − 1.97e4i)17-s + ⋯ |
| L(s) = 1 | + (1.21 + 0.783i)2-s + (−0.254 − 1.76i)3-s + (0.456 + 1.00i)4-s + (−1.16 − 0.341i)5-s + (1.07 − 2.35i)6-s + (0.531 − 0.613i)7-s + (−0.0205 + 0.142i)8-s + (−2.09 + 0.616i)9-s + (−1.14 − 1.32i)10-s + (0.611 − 0.393i)11-s + (1.65 − 1.06i)12-s + (0.750 + 0.865i)13-s + (1.12 − 0.331i)14-s + (−0.307 + 2.14i)15-s + (0.583 − 0.673i)16-s + (0.444 − 0.972i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.75637 - 1.52409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75637 - 1.52409i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (5.66e4 - 1.40e4i)T \) |
| good | 2 | \( 1 + (-13.7 - 8.86i)T + (53.1 + 116. i)T^{2} \) |
| 3 | \( 1 + (11.8 + 82.6i)T + (-2.09e3 + 616. i)T^{2} \) |
| 5 | \( 1 + (324. + 95.3i)T + (6.57e4 + 4.22e4i)T^{2} \) |
| 7 | \( 1 + (-482. + 556. i)T + (-1.17e5 - 8.15e5i)T^{2} \) |
| 11 | \( 1 + (-2.70e3 + 1.73e3i)T + (8.09e6 - 1.77e7i)T^{2} \) |
| 13 | \( 1 + (-5.94e3 - 6.85e3i)T + (-8.93e6 + 6.21e7i)T^{2} \) |
| 17 | \( 1 + (-8.99e3 + 1.97e4i)T + (-2.68e8 - 3.10e8i)T^{2} \) |
| 19 | \( 1 + (-1.72e4 - 3.77e4i)T + (-5.85e8 + 6.75e8i)T^{2} \) |
| 29 | \( 1 + (-7.73e4 + 1.69e5i)T + (-1.12e10 - 1.30e10i)T^{2} \) |
| 31 | \( 1 + (-3.83e3 + 2.66e4i)T + (-2.63e10 - 7.75e9i)T^{2} \) |
| 37 | \( 1 + (3.21e5 - 9.44e4i)T + (7.98e10 - 5.13e10i)T^{2} \) |
| 41 | \( 1 + (-2.71e5 - 7.96e4i)T + (1.63e11 + 1.05e11i)T^{2} \) |
| 43 | \( 1 + (3.35e4 + 2.33e5i)T + (-2.60e11 + 7.65e10i)T^{2} \) |
| 47 | \( 1 - 1.13e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (4.71e4 - 5.44e4i)T + (-1.67e11 - 1.16e12i)T^{2} \) |
| 59 | \( 1 + (-1.23e6 - 1.42e6i)T + (-3.54e11 + 2.46e12i)T^{2} \) |
| 61 | \( 1 + (36.7 - 255. i)T + (-3.01e12 - 8.85e11i)T^{2} \) |
| 67 | \( 1 + (2.83e6 + 1.81e6i)T + (2.51e12 + 5.51e12i)T^{2} \) |
| 71 | \( 1 + (-3.20e6 - 2.05e6i)T + (3.77e12 + 8.27e12i)T^{2} \) |
| 73 | \( 1 + (7.07e5 + 1.54e6i)T + (-7.23e12 + 8.34e12i)T^{2} \) |
| 79 | \( 1 + (-9.79e5 - 1.13e6i)T + (-2.73e12 + 1.90e13i)T^{2} \) |
| 83 | \( 1 + (8.23e6 - 2.41e6i)T + (2.28e13 - 1.46e13i)T^{2} \) |
| 89 | \( 1 + (-2.49e5 - 1.73e6i)T + (-4.24e13 + 1.24e13i)T^{2} \) |
| 97 | \( 1 + (-3.36e6 - 9.86e5i)T + (6.79e13 + 4.36e13i)T^{2} \) |
| 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
−15.94376248310142871404608259899, −14.07755493378563566740229156984, −13.77702741357607242068582401680, −12.13160282644945996155685365054, −11.72469838005595706356447201623, −8.062941255027325840840186440861, −7.23163571580569931826732964334, −5.93570322902267179944614760198, −3.99227319567724336489101273447, −0.981524374336012886596041779668,
3.28162162951651285920631373479, 4.23072543707188580111065243184, 5.47937000219620821221270094790, 8.571429639244069445430158882837, 10.50418551456200920416144370422, 11.32838523894282485716709339865, 12.24759484041071144873943072664, 14.34424236311161800874624410133, 15.19226210564532656448660248785, 15.87088315560327814099619173987
