| L(s) = 1 | + (10.9 + 13.1i)3-s + (−41.8 − 15.2i)5-s + (−26.2 + 45.5i)7-s + (−36.7 + 208. i)9-s + (−55.1 − 95.5i)11-s + (31.8 − 37.9i)13-s + (−260. − 716. i)15-s + (62.4 + 354. i)17-s + (346. + 100. i)19-s + (−885. + 156. i)21-s + (−253. + 92.2i)23-s + (1.04e3 + 875. i)25-s + (−1.93e3 + 1.11e3i)27-s + (−563. − 99.3i)29-s + (910. + 525. i)31-s + ⋯ |
| L(s) = 1 | + (1.22 + 1.45i)3-s + (−1.67 − 0.609i)5-s + (−0.536 + 0.928i)7-s + (−0.454 + 2.57i)9-s + (−0.455 − 0.789i)11-s + (0.188 − 0.224i)13-s + (−1.15 − 3.18i)15-s + (0.216 + 1.22i)17-s + (0.960 + 0.278i)19-s + (−2.00 + 0.354i)21-s + (−0.479 + 0.174i)23-s + (1.67 + 1.40i)25-s + (−2.65 + 1.53i)27-s + (−0.670 − 0.118i)29-s + (0.947 + 0.546i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.274683 + 1.24684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.274683 + 1.24684i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-346. - 100. i)T \) |
| good | 3 | \( 1 + (-10.9 - 13.1i)T + (-14.0 + 79.7i)T^{2} \) |
| 5 | \( 1 + (41.8 + 15.2i)T + (478. + 401. i)T^{2} \) |
| 7 | \( 1 + (26.2 - 45.5i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (55.1 + 95.5i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-31.8 + 37.9i)T + (-4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-62.4 - 354. i)T + (-7.84e4 + 2.85e4i)T^{2} \) |
| 23 | \( 1 + (253. - 92.2i)T + (2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (563. + 99.3i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-910. - 525. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 527. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-407. - 485. i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (910. + 331. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (22.9 - 130. i)T + (-4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + (203. + 558. i)T + (-6.04e6 + 5.07e6i)T^{2} \) |
| 59 | \( 1 + (-3.75e3 + 661. i)T + (1.13e7 - 4.14e6i)T^{2} \) |
| 61 | \( 1 + (-1.59e3 + 579. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-2.49e3 - 439. i)T + (1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-1.82e3 + 5.02e3i)T + (-1.94e7 - 1.63e7i)T^{2} \) |
| 73 | \( 1 + (-500. + 419. i)T + (4.93e6 - 2.79e7i)T^{2} \) |
| 79 | \( 1 + (53.2 + 63.4i)T + (-6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (3.32e3 - 5.76e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-3.75e3 + 4.47e3i)T + (-1.08e7 - 6.17e7i)T^{2} \) |
| 97 | \( 1 + (1.42e4 - 2.50e3i)T + (8.31e7 - 3.02e7i)T^{2} \) |
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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
−14.62275011930843334476801184088, −13.24793145424029980221934685147, −11.97134761175377282653214772922, −10.79204165016851341464400036299, −9.536243850471069217059952663189, −8.410630484739021715139885334327, −8.050219036601814330113364256103, −5.33647417514499620562513131876, −3.92979211386820825047526213536, −3.12080936508121241949160742968,
0.56819780491625099908756089615, 2.79434061490788684389114294930, 3.89901175396619903667826060180, 6.97905395342475437858157649020, 7.31351120122376137678790815871, 8.201168292422202187284544492775, 9.717241586111556398804162336640, 11.46720406936709270609475702389, 12.31004085278450161366971804698, 13.38878851590591423536143639102
