| L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.558 − 3.16i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (0.558 − 3.16i)6-s + (0.0116 − 0.0202i)7-s + (0.500 + 0.866i)8-s + (−6.89 + 2.51i)9-s + (0.939 − 0.342i)10-s + (−1.08 − 1.87i)11-s + (1.60 − 2.78i)12-s + (−0.276 + 1.56i)13-s + (0.0179 − 0.0150i)14-s + (−2.46 − 2.06i)15-s + (0.173 + 0.984i)16-s + (7.49 + 2.72i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.322 − 1.82i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (0.227 − 1.29i)6-s + (0.00442 − 0.00765i)7-s + (0.176 + 0.306i)8-s + (−2.29 + 0.836i)9-s + (0.297 − 0.108i)10-s + (−0.325 − 0.564i)11-s + (0.464 − 0.803i)12-s + (−0.0765 + 0.434i)13-s + (0.00478 − 0.00401i)14-s + (−0.636 − 0.533i)15-s + (0.0434 + 0.246i)16-s + (1.81 + 0.661i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22095 - 0.960266i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22095 - 0.960266i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-1.06 + 4.22i)T \) |
| good | 3 | \( 1 + (0.558 + 3.16i)T + (-2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (-0.0116 + 0.0202i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.08 + 1.87i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.276 - 1.56i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-7.49 - 2.72i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.43 - 2.88i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (7.09 - 2.58i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.22 - 3.86i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.389T + 37T^{2} \) |
| 41 | \( 1 + (-0.972 - 5.51i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.43 + 6.23i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (4.01 - 1.46i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (2.89 + 2.43i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.41 - 1.24i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (8.84 + 7.41i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.62 - 1.31i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (9.85 - 8.27i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.330 + 1.87i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.65 + 9.36i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.31 + 2.27i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.54 + 8.74i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (10.4 + 3.81i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.61143355836178911918931654479, −11.75524516009922058033241462108, −10.85773763566832484602121808056, −9.031930978365775595658178582241, −7.83089003174240823550031975699, −7.14807799854427395880965654405, −5.98660439237550053426189062065, −5.30829184557074324958913404097, −3.06420170532237962167805788610, −1.45513449057481048142848604328,
2.95371432639306828263201144443, 4.01052783924476021343912803386, 5.24882316009040232360915824759, 5.81687406050195603012543903052, 7.63940053569355466404016394059, 9.329110423072488254430038023115, 10.01496960548138457804207449532, 10.66635988990291927735530107585, 11.63935845648922816091608050199, 12.60009826353248086658153214265
