| L(s) = 1 | + (−4.54 − 0.720i)2-s + (0.590 + 1.15i)3-s + (4.94 + 1.60i)4-s + (22.6 + 10.6i)5-s + (−1.85 − 5.69i)6-s + (49.8 + 49.8i)7-s + (44.3 + 22.5i)8-s + (46.6 − 64.1i)9-s + (−95.1 − 64.7i)10-s + (−143. + 103. i)11-s + (1.05 + 6.67i)12-s + (67.0 − 10.6i)13-s + (−190. − 262. i)14-s + (1.01 + 32.4i)15-s + (−252. − 183. i)16-s + (−66.5 + 130. i)17-s + ⋯ |
| L(s) = 1 | + (−1.13 − 0.180i)2-s + (0.0655 + 0.128i)3-s + (0.308 + 0.100i)4-s + (0.904 + 0.425i)5-s + (−0.0513 − 0.158i)6-s + (1.01 + 1.01i)7-s + (0.692 + 0.352i)8-s + (0.575 − 0.792i)9-s + (−0.951 − 0.647i)10-s + (−1.18 + 0.858i)11-s + (0.00734 + 0.0463i)12-s + (0.396 − 0.0628i)13-s + (−0.973 − 1.33i)14-s + (0.00451 + 0.144i)15-s + (−0.986 − 0.716i)16-s + (−0.230 + 0.451i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.875024 + 0.224545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.875024 + 0.224545i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-22.6 - 10.6i)T \) |
| good | 2 | \( 1 + (4.54 + 0.720i)T + (15.2 + 4.94i)T^{2} \) |
| 3 | \( 1 + (-0.590 - 1.15i)T + (-47.6 + 65.5i)T^{2} \) |
| 7 | \( 1 + (-49.8 - 49.8i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + (143. - 103. i)T + (4.52e3 - 1.39e4i)T^{2} \) |
| 13 | \( 1 + (-67.0 + 10.6i)T + (2.71e4 - 8.82e3i)T^{2} \) |
| 17 | \( 1 + (66.5 - 130. i)T + (-4.90e4 - 6.75e4i)T^{2} \) |
| 19 | \( 1 + (-497. + 161. i)T + (1.05e5 - 7.66e4i)T^{2} \) |
| 23 | \( 1 + (36.0 - 227. i)T + (-2.66e5 - 8.64e4i)T^{2} \) |
| 29 | \( 1 + (908. + 295. i)T + (5.72e5 + 4.15e5i)T^{2} \) |
| 31 | \( 1 + (163. + 501. i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (340. + 2.15e3i)T + (-1.78e6 + 5.79e5i)T^{2} \) |
| 41 | \( 1 + (260. + 189. i)T + (8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 + (167. - 167. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (1.10e3 - 563. i)T + (2.86e6 - 3.94e6i)T^{2} \) |
| 53 | \( 1 + (804. + 1.57e3i)T + (-4.63e6 + 6.38e6i)T^{2} \) |
| 59 | \( 1 + (-3.66e3 + 5.03e3i)T + (-3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (14.9 - 10.8i)T + (4.27e6 - 1.31e7i)T^{2} \) |
| 67 | \( 1 + (3.03e3 - 5.96e3i)T + (-1.18e7 - 1.63e7i)T^{2} \) |
| 71 | \( 1 + (958. - 2.94e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (-1.20e3 + 7.59e3i)T + (-2.70e7 - 8.77e6i)T^{2} \) |
| 79 | \( 1 + (3.08e3 + 1.00e3i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-4.38e3 - 2.23e3i)T + (2.78e7 + 3.83e7i)T^{2} \) |
| 89 | \( 1 + (4.86e3 + 6.69e3i)T + (-1.93e7 + 5.96e7i)T^{2} \) |
| 97 | \( 1 + (-2.72e3 + 1.39e3i)T + (5.20e7 - 7.16e7i)T^{2} \) |
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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
−17.70523588531204842865594557869, −15.73548949325682682937635928440, −14.62741386958424488280329810163, −13.08437223346101168983055374430, −11.30496000673076685334928352928, −9.997900058939366599175933084258, −9.083628474153692603917070584142, −7.54857037940882837452744088187, −5.32267347583491793778414490897, −1.93942155140623865865650806445,
1.30503364560112321939805899141, 4.99881527174906956830612910153, 7.44063683755344873639418734406, 8.432791214381131634082733271924, 10.02809949526198808035011102916, 10.90040817290900633260440543333, 13.34819785820051140812207221715, 13.83402561050357159538360248608, 16.14193298815318392320365562080, 16.81889286557315108332070512423
