| L(s) = 1 | + (−3.79 + 1.25i)2-s + (−8.69 − 3.97i)3-s + (12.8 − 9.55i)4-s + (17.8 − 20.5i)5-s + (38.0 + 4.13i)6-s + (−0.355 − 0.553i)7-s + (−36.6 + 52.4i)8-s + (6.83 + 7.88i)9-s + (−41.8 + 100. i)10-s + (177. + 25.5i)11-s + (−149. + 32.1i)12-s + (85.7 + 55.1i)13-s + (2.04 + 1.65i)14-s + (−236. + 108. i)15-s + (73.2 − 245. i)16-s + (−456. − 134. i)17-s + ⋯ |
| L(s) = 1 | + (−0.949 + 0.314i)2-s + (−0.966 − 0.441i)3-s + (0.801 − 0.597i)4-s + (0.713 − 0.823i)5-s + (1.05 + 0.114i)6-s + (−0.00726 − 0.0112i)7-s + (−0.573 + 0.819i)8-s + (0.0844 + 0.0974i)9-s + (−0.418 + 1.00i)10-s + (1.46 + 0.211i)11-s + (−1.03 + 0.223i)12-s + (0.507 + 0.326i)13-s + (0.0104 + 0.00844i)14-s + (−1.05 + 0.480i)15-s + (0.286 − 0.958i)16-s + (−1.58 − 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.221767 - 0.541439i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.221767 - 0.541439i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.79 - 1.25i)T \) |
| 23 | \( 1 + (164. + 502. i)T \) |
| good | 3 | \( 1 + (8.69 + 3.97i)T + (53.0 + 61.2i)T^{2} \) |
| 5 | \( 1 + (-17.8 + 20.5i)T + (-88.9 - 618. i)T^{2} \) |
| 7 | \( 1 + (0.355 + 0.553i)T + (-997. + 2.18e3i)T^{2} \) |
| 11 | \( 1 + (-177. - 25.5i)T + (1.40e4 + 4.12e3i)T^{2} \) |
| 13 | \( 1 + (-85.7 - 55.1i)T + (1.18e4 + 2.59e4i)T^{2} \) |
| 17 | \( 1 + (456. + 134. i)T + (7.02e4 + 4.51e4i)T^{2} \) |
| 19 | \( 1 + (173. + 589. i)T + (-1.09e5 + 7.04e4i)T^{2} \) |
| 29 | \( 1 + (144. + 42.3i)T + (5.95e5 + 3.82e5i)T^{2} \) |
| 31 | \( 1 + (449. - 205. i)T + (6.04e5 - 6.97e5i)T^{2} \) |
| 37 | \( 1 + (900. + 1.03e3i)T + (-2.66e5 + 1.85e6i)T^{2} \) |
| 41 | \( 1 + (808. - 932. i)T + (-4.02e5 - 2.79e6i)T^{2} \) |
| 43 | \( 1 + (1.10e3 + 503. i)T + (2.23e6 + 2.58e6i)T^{2} \) |
| 47 | \( 1 - 1.00e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (3.69e3 - 2.37e3i)T + (3.27e6 - 7.17e6i)T^{2} \) |
| 59 | \( 1 + (725. - 1.12e3i)T + (-5.03e6 - 1.10e7i)T^{2} \) |
| 61 | \( 1 + (2.14e3 + 4.69e3i)T + (-9.06e6 + 1.04e7i)T^{2} \) |
| 67 | \( 1 + (-1.58e3 + 227. i)T + (1.93e7 - 5.67e6i)T^{2} \) |
| 71 | \( 1 + (-857. + 123. i)T + (2.43e7 - 7.15e6i)T^{2} \) |
| 73 | \( 1 + (-6.65e3 + 1.95e3i)T + (2.38e7 - 1.53e7i)T^{2} \) |
| 79 | \( 1 + (-332. + 517. i)T + (-1.61e7 - 3.54e7i)T^{2} \) |
| 83 | \( 1 + (-2.57e3 + 2.23e3i)T + (6.75e6 - 4.69e7i)T^{2} \) |
| 89 | \( 1 + (-1.20e3 + 2.63e3i)T + (-4.10e7 - 4.74e7i)T^{2} \) |
| 97 | \( 1 + (-1.10e4 + 1.27e4i)T + (-1.25e7 - 8.76e7i)T^{2} \) |
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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
−12.73042667944853574914590494402, −11.60916260527254169338101213556, −10.89887969980788542452644225239, −9.190946092585625429516077471030, −8.898978837643890486721765874683, −6.80920471507405967944208751297, −6.33894394750368536690224914943, −4.88412724978333978545728520484, −1.77398196353872533657973743346, −0.42291462041935654264010150055,
1.78473468759218149577968049631, 3.76538602191218221919035742094, 6.00528807364239584142772399230, 6.62302912941323146446395307426, 8.389388521777813804052945328320, 9.634693761968830788984565474574, 10.58348884212468742096862963508, 11.21260541252737435261485378835, 12.18512103300789125244681301803, 13.70753498770678866455003411210
