| L(s) = 1 | + (−2.68 − 0.874i)2-s + (4.50 − 3.27i)3-s + (6.47 + 4.70i)4-s + (−10.0 − 30.8i)5-s + (−14.9 + 4.87i)6-s + (50.5 − 69.5i)7-s + (−13.3 − 18.3i)8-s + (−15.4 + 47.4i)9-s + 91.6i·10-s + (−44.2 + 112. i)11-s + 44.5·12-s + (97.9 + 31.8i)13-s + (−196. + 142. i)14-s + (−146. − 106. i)15-s + (19.7 + 60.8i)16-s + (121. − 39.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 − 0.218i)2-s + (0.501 − 0.364i)3-s + (0.404 + 0.293i)4-s + (−0.400 − 1.23i)5-s + (−0.416 + 0.135i)6-s + (1.03 − 1.41i)7-s + (−0.207 − 0.286i)8-s + (−0.190 + 0.586i)9-s + 0.916i·10-s + (−0.365 + 0.930i)11-s + 0.309·12-s + (0.579 + 0.188i)13-s + (−1.00 + 0.728i)14-s + (−0.649 − 0.471i)15-s + (0.0772 + 0.237i)16-s + (0.418 − 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.858061 - 0.679218i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.858061 - 0.679218i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.68 + 0.874i)T \) |
| 11 | \( 1 + (44.2 - 112. i)T \) |
| good | 3 | \( 1 + (-4.50 + 3.27i)T + (25.0 - 77.0i)T^{2} \) |
| 5 | \( 1 + (10.0 + 30.8i)T + (-505. + 367. i)T^{2} \) |
| 7 | \( 1 + (-50.5 + 69.5i)T + (-741. - 2.28e3i)T^{2} \) |
| 13 | \( 1 + (-97.9 - 31.8i)T + (2.31e4 + 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-121. + 39.3i)T + (6.75e4 - 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-158. - 217. i)T + (-4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 + 446.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-639. + 879. i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 31 | \( 1 + (155. - 478. i)T + (-7.47e5 - 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-1.56e3 - 1.13e3i)T + (5.79e5 + 1.78e6i)T^{2} \) |
| 41 | \( 1 + (-951. - 1.30e3i)T + (-8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 + 1.44e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (2.66e3 - 1.93e3i)T + (1.50e6 - 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-906. + 2.79e3i)T + (-6.38e6 - 4.63e6i)T^{2} \) |
| 59 | \( 1 + (1.30e3 + 947. i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (3.17e3 - 1.03e3i)T + (1.12e7 - 8.13e6i)T^{2} \) |
| 67 | \( 1 + 2.14e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-1.04e3 - 3.21e3i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (4.44e3 - 6.11e3i)T + (-8.77e6 - 2.70e7i)T^{2} \) |
| 79 | \( 1 + (6.15e3 + 2.00e3i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-3.86e3 + 1.25e3i)T + (3.83e7 - 2.78e7i)T^{2} \) |
| 89 | \( 1 + 6.03e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-4.24e3 + 1.30e4i)T + (-7.16e7 - 5.20e7i)T^{2} \) |
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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
−17.01778786226178482409087851614, −16.12335347744850094738368317655, −14.24625825233194609817275709804, −13.05794711540187654696665921141, −11.59516172528070314349811314645, −10.10092915325940225140256119963, −8.263876430913609732258670092434, −7.65456134968611621343024494820, −4.52607543903145141929038022420, −1.35965912786835398253891122184,
2.93675260341134483727610053522, 5.97694256821375657255943652194, 7.962673432231501763218339075063, 9.041853493960163527348929064967, 10.80472426725800272143635595110, 11.79338225758037145433097337420, 14.30027533784284352573411823347, 15.05626173528715206040735458212, 15.94415046293114665736247510734, 18.09453443127283532777021421118
