| L(s) = 1 | + (−0.699 + 1.92i)2-s + (−4.26 + 1.55i)3-s + (2.92 + 2.45i)4-s + (−16.6 − 2.92i)5-s − 9.28i·6-s + (0.336 − 1.90i)7-s + (−20.9 + 12.0i)8-s + (−4.89 + 4.10i)9-s + (17.2 − 29.8i)10-s + (20.7 + 35.9i)11-s + (−16.2 − 5.93i)12-s + (16.9 − 20.1i)13-s + (3.42 + 1.97i)14-s + (75.3 − 13.2i)15-s + (−3.27 − 18.5i)16-s + (51.5 + 61.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.247 + 0.679i)2-s + (−0.821 + 0.298i)3-s + (0.365 + 0.306i)4-s + (−1.48 − 0.261i)5-s − 0.631i·6-s + (0.0181 − 0.102i)7-s + (−0.924 + 0.534i)8-s + (−0.181 + 0.151i)9-s + (0.545 − 0.944i)10-s + (0.569 + 0.985i)11-s + (−0.392 − 0.142i)12-s + (0.361 − 0.430i)13-s + (0.0654 + 0.0377i)14-s + (1.29 − 0.228i)15-s + (−0.0511 − 0.290i)16-s + (0.735 + 0.876i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0363382 + 0.522140i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0363382 + 0.522140i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (139. - 176. i)T \) |
| good | 2 | \( 1 + (0.699 - 1.92i)T + (-6.12 - 5.14i)T^{2} \) |
| 3 | \( 1 + (4.26 - 1.55i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (16.6 + 2.92i)T + (117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-0.336 + 1.90i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-20.7 - 35.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-16.9 + 20.1i)T + (-381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-51.5 - 61.4i)T + (-853. + 4.83e3i)T^{2} \) |
| 19 | \( 1 + (-30.0 - 82.6i)T + (-5.25e3 + 4.40e3i)T^{2} \) |
| 23 | \( 1 + (148. + 85.5i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (112. - 64.9i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 64.5iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-213. - 179. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + 60.7iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-282. + 489. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-79.1 - 448. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (14.8 - 2.61i)T + (1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (228. - 272. i)T + (-3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (62.6 - 355. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (434. - 157. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + 240.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-538. - 95.0i)T + (4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (686. - 575. i)T + (9.92e4 - 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-1.28e3 + 227. i)T + (6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (432. + 249. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−16.49238534455461756185232598638, −15.59929555656160203144610716784, −14.65609663236904667447457205343, −12.25685599127155901280226952038, −11.86451921094190781914045682772, −10.48284711550870815495472591295, −8.381208428590897460772740769277, −7.48540610773200932333773721728, −5.88156429312581036330887030689, −3.98766853350042313126872441670,
0.53265276615969107172925203851, 3.47329165650880479329767699943, 5.91770810755034594477282481281, 7.31162357523622292176785644986, 9.118422182196997622876448804688, 10.97286224215805617306644542482, 11.59205861114499152239875535356, 12.13663242419162646479489909669, 14.19937310064115580246828485408, 15.61352314380394397700559572343
