| L(s) = 1 | + (−7.77 − 9.26i)3-s + (42.9 + 15.6i)5-s + (2.15 − 3.72i)7-s + (−11.3 + 64.1i)9-s + (−53.2 − 92.2i)11-s + (157. − 187. i)13-s + (−188. − 519. i)15-s + (−15.1 − 85.6i)17-s + (−115. − 341. i)19-s + (−51.2 + 9.03i)21-s + (55.9 − 20.3i)23-s + (1.12e3 + 940. i)25-s + (−165. + 95.6i)27-s + (−226. − 39.9i)29-s + (581. + 335. i)31-s + ⋯ |
| L(s) = 1 | + (−0.863 − 1.02i)3-s + (1.71 + 0.625i)5-s + (0.0439 − 0.0760i)7-s + (−0.139 + 0.792i)9-s + (−0.440 − 0.762i)11-s + (0.930 − 1.10i)13-s + (−0.839 − 2.30i)15-s + (−0.0522 − 0.296i)17-s + (−0.321 − 0.946i)19-s + (−0.116 + 0.0204i)21-s + (0.105 − 0.0384i)23-s + (1.79 + 1.50i)25-s + (−0.227 + 0.131i)27-s + (−0.269 − 0.0475i)29-s + (0.605 + 0.349i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.20726 - 1.01175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20726 - 1.01175i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (115. + 341. i)T \) |
| good | 3 | \( 1 + (7.77 + 9.26i)T + (-14.0 + 79.7i)T^{2} \) |
| 5 | \( 1 + (-42.9 - 15.6i)T + (478. + 401. i)T^{2} \) |
| 7 | \( 1 + (-2.15 + 3.72i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (53.2 + 92.2i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-157. + 187. i)T + (-4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (15.1 + 85.6i)T + (-7.84e4 + 2.85e4i)T^{2} \) |
| 23 | \( 1 + (-55.9 + 20.3i)T + (2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (226. + 39.9i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-581. - 335. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 2.47e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (650. + 775. i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-2.56e3 - 933. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (270. - 1.53e3i)T + (-4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + (-1.04e3 - 2.88e3i)T + (-6.04e6 + 5.07e6i)T^{2} \) |
| 59 | \( 1 + (-4.89e3 + 863. i)T + (1.13e7 - 4.14e6i)T^{2} \) |
| 61 | \( 1 + (5.54e3 - 2.01e3i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-698. - 123. i)T + (1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (149. - 410. i)T + (-1.94e7 - 1.63e7i)T^{2} \) |
| 73 | \( 1 + (4.82e3 - 4.04e3i)T + (4.93e6 - 2.79e7i)T^{2} \) |
| 79 | \( 1 + (5.31e3 + 6.33e3i)T + (-6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (6.60e3 - 1.14e4i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-5.16e3 + 6.15e3i)T + (-1.08e7 - 6.17e7i)T^{2} \) |
| 97 | \( 1 + (1.44e4 - 2.54e3i)T + (8.31e7 - 3.02e7i)T^{2} \) |
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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
−13.35238425779316398398666626573, −12.76286859885953745223126576099, −11.11476984299296006065071555694, −10.51986125521089397175355422802, −9.011299134001377916448411601126, −7.31063372856775975394393027941, −6.11293976390144396908193213716, −5.60251495156157874617930759934, −2.64595660958368098521746004548, −0.974122486419539638590609935284,
1.78273370006216372221194253622, 4.39265380950616881917736936146, 5.47050013051377300039882132543, 6.38875680909264235423243883532, 8.697279200505890641436724100062, 9.831365063735159253568261080544, 10.35145663404171218763456301854, 11.69104267597451895917387115026, 12.99747428767959604666861423619, 13.90586170941147065399813615893
