| L(s) = 1 | + (−7.81 − 10.7i)2-s + (37.7 + 12.2i)3-s + (−15.0 + 46.4i)4-s + (−244. + 135. i)5-s + (−163. − 501. i)6-s − 876. i·7-s + (−1.00e3 + 325. i)8-s + (−495. − 359. i)9-s + (3.37e3 + 1.56e3i)10-s + (−1.88e3 + 1.36e3i)11-s + (−1.13e3 + 1.56e3i)12-s + (−7.81e3 + 1.07e4i)13-s + (−9.42e3 + 6.84e3i)14-s + (−1.08e4 + 2.13e3i)15-s + (1.63e4 + 1.19e4i)16-s + (583. − 189. i)17-s + ⋯ |
| L(s) = 1 | + (−0.690 − 0.950i)2-s + (0.807 + 0.262i)3-s + (−0.117 + 0.362i)4-s + (−0.873 + 0.486i)5-s + (−0.308 − 0.948i)6-s − 0.965i·7-s + (−0.691 + 0.224i)8-s + (−0.226 − 0.164i)9-s + (1.06 + 0.494i)10-s + (−0.426 + 0.309i)11-s + (−0.190 + 0.261i)12-s + (−0.986 + 1.35i)13-s + (−0.918 + 0.667i)14-s + (−0.832 + 0.163i)15-s + (0.999 + 0.726i)16-s + (0.0288 − 0.00936i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0683967 + 0.175413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0683967 + 0.175413i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (244. - 135. i)T \) |
| good | 2 | \( 1 + (7.81 + 10.7i)T + (-39.5 + 121. i)T^{2} \) |
| 3 | \( 1 + (-37.7 - 12.2i)T + (1.76e3 + 1.28e3i)T^{2} \) |
| 7 | \( 1 + 876. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + (1.88e3 - 1.36e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (7.81e3 - 1.07e4i)T + (-1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-583. + 189. i)T + (3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (4.03e3 + 1.24e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (4.43e4 + 6.10e4i)T + (-1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (6.14e4 - 1.89e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (9.72e3 + 2.99e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.21e5 + 3.04e5i)T + (-2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-1.06e5 - 7.72e4i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 8.90e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + (8.08e5 + 2.62e5i)T + (4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-8.16e5 - 2.65e5i)T + (9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-4.25e5 - 3.09e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (8.62e4 - 6.26e4i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (1.75e6 - 5.69e5i)T + (4.90e12 - 3.56e12i)T^{2} \) |
| 71 | \( 1 + (8.23e5 - 2.53e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.45e6 - 2.00e6i)T + (-3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-7.68e5 + 2.36e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (1.12e6 - 3.64e5i)T + (2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + (8.43e6 - 6.13e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (1.09e7 + 3.55e6i)T + (6.53e13 + 4.74e13i)T^{2} \) |
| 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
−14.97427060193567974129290598870, −14.21577549304998514460640305612, −12.22559985502951848190528611132, −11.05388464915707623362954044963, −9.935883003564508716461262005386, −8.675096961127436105288888601076, −7.14379150138648945535139803504, −3.99667410564287749083790509806, −2.44846501333908046719977938750, −0.10354025722071914133299739480,
2.94963200132776992690694977060, 5.61495865904723084611332258841, 7.85927259338384468422441568865, 8.100931148206402498026131082934, 9.502187222989669116855803623791, 11.79956200922542598612144807872, 12.99556274682502163028374838544, 14.87613943404700532355559524665, 15.47890914555145489918004288943, 16.63706480052599024269531488123
