| L(s) = 1 | + (−5.09 − 0.267i)2-s + (−3.82 − 4.72i)3-s + (17.9 + 1.88i)4-s + (3.95 + 10.4i)5-s + (18.2 + 25.1i)6-s + (−12.8 + 13.3i)7-s + (−50.7 − 8.03i)8-s + (−2.06 + 9.71i)9-s + (−17.3 − 54.3i)10-s + (21.9 − 4.65i)11-s + (−59.7 − 92.0i)12-s + (33.3 + 16.9i)13-s + (69.0 − 64.6i)14-s + (34.2 − 58.6i)15-s + (115. + 24.4i)16-s + (−29.1 − 75.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.80 − 0.0944i)2-s + (−0.736 − 0.908i)3-s + (2.24 + 0.236i)4-s + (0.353 + 0.935i)5-s + (1.24 + 1.70i)6-s + (−0.693 + 0.720i)7-s + (−2.24 − 0.355i)8-s + (−0.0765 + 0.359i)9-s + (−0.549 − 1.71i)10-s + (0.600 − 0.127i)11-s + (−1.43 − 2.21i)12-s + (0.711 + 0.362i)13-s + (1.31 − 1.23i)14-s + (0.589 − 1.01i)15-s + (1.80 + 0.382i)16-s + (−0.416 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0101757 + 0.0453868i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0101757 + 0.0453868i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-3.95 - 10.4i)T \) |
| 7 | \( 1 + (12.8 - 13.3i)T \) |
| good | 2 | \( 1 + (5.09 + 0.267i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (3.82 + 4.72i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (-21.9 + 4.65i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-33.3 - 16.9i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (29.1 + 75.9i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (-3.31 - 31.5i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-6.55 + 124. i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (55.2 - 76.0i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-40.3 + 90.5i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (273. - 177. i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (32.4 - 10.5i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-359. - 359. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (373. + 143. i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (164. - 133. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (169. - 187. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (630. - 567. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (-299. + 115. i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (541. + 393. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (161. - 249. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (371. + 834. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-224. + 1.41e3i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (375. + 417. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (-4.41 - 27.8i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−12.11544210014523139448989464335, −11.51593119728682573402470197407, −10.64719692850452112188614083364, −9.562970897477419168043126532404, −8.816944439013101209948807898710, −7.42063601517254410246112862559, −6.55461339287343896897920120630, −6.13015117302834241932422948995, −2.89568429400487059610575921109, −1.53730171845145895952474750975,
0.04387518902992628005072143432, 1.47599874178453165424476038880, 3.95308734060965433897802337235, 5.62661867928544391276441642574, 6.67454787416918068318559159111, 8.010225229947765486887841159359, 9.101070034993454143601709931932, 9.717908133230563190271249787399, 10.58048050950233108331980518822, 11.20198694664745105408986535760
