| L(s) = 1 | + (−8.74 + 2.84i)2-s + (16.5 + 22.7i)3-s + (42.5 − 30.9i)4-s + (53.9 + 14.5i)5-s + (−209. − 152. i)6-s + 122. i·7-s + (−111. + 153. i)8-s + (−170. + 523. i)9-s + (−513. + 25.7i)10-s + (−164. − 505. i)11-s + (1.40e3 + 457. i)12-s + (−207. − 67.4i)13-s + (−349. − 1.07e3i)14-s + (560. + 1.47e3i)15-s + (18.2 − 56.2i)16-s + (527. − 725. i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 + 0.502i)2-s + (1.06 + 1.46i)3-s + (1.32 − 0.966i)4-s + (0.965 + 0.261i)5-s + (−2.37 − 1.72i)6-s + 0.948i·7-s + (−0.615 + 0.846i)8-s + (−0.700 + 2.15i)9-s + (−1.62 + 0.0813i)10-s + (−0.409 − 1.25i)11-s + (2.82 + 0.917i)12-s + (−0.340 − 0.110i)13-s + (−0.476 − 1.46i)14-s + (0.643 + 1.68i)15-s + (0.0178 − 0.0549i)16-s + (0.442 − 0.609i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.405316 + 0.986715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.405316 + 0.986715i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-53.9 - 14.5i)T \) |
| good | 2 | \( 1 + (8.74 - 2.84i)T + (25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (-16.5 - 22.7i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 122. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (164. + 505. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (207. + 67.4i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-527. + 725. i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-530. - 385. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (574. - 186. i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.68e3 + 1.22e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (2.20e3 + 1.60e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.01e4 - 3.28e3i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-2.53e3 + 7.81e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 7.20e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.25e3 - 1.72e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.30e4 - 1.79e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-2.55e3 + 7.86e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.52e4 + 4.68e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.80e4 - 5.23e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-4.38e4 + 3.18e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-3.03e4 + 9.86e3i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (2.47e4 - 1.79e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-1.47e4 + 2.03e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.37e4 - 7.31e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (8.62e4 + 1.18e5i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.76834374417716340340762775106, −15.97584261591907979947436689372, −14.98066017602596283157146813017, −13.81833491174065346740058213009, −10.90691032016868046094814788604, −9.802766133962944665271046147262, −9.126303495470230890653147927229, −8.062051595701968258567286978326, −5.66610784256890933124468914783, −2.69175059412678595688046403601,
1.16626069503748539246137170966, 2.34664132410803581592579573353, 6.97550783129229392923106792866, 7.88838769881315862276575650061, 9.227878143751679468820400651521, 10.26597475862134346943122047274, 12.28813364731675405637879175044, 13.34407496538728126500037014154, 14.54257920771944506805915700633, 16.84133626789624829516354712404
