| L(s) = 1 | + (0.00173 − 1.41i)2-s + (−1.61 + 2.30i)3-s + (−1.99 − 0.00489i)4-s + (−2.13 + 0.654i)5-s + (3.25 + 2.28i)6-s + (0.0970 + 0.362i)7-s + (−0.0103 + 2.82i)8-s + (−1.68 − 4.61i)9-s + (0.922 + 3.02i)10-s + (3.54 − 2.04i)11-s + (3.23 − 4.60i)12-s + (−2.59 − 3.70i)13-s + (0.512 − 0.136i)14-s + (1.94 − 5.98i)15-s + (3.99 + 0.0195i)16-s + (−2.42 − 5.19i)17-s + ⋯ |
| L(s) = 1 | + (0.00122 − 0.999i)2-s + (−0.931 + 1.33i)3-s + (−0.999 − 0.00244i)4-s + (−0.956 + 0.292i)5-s + (1.32 + 0.933i)6-s + (0.0366 + 0.136i)7-s + (−0.00367 + 0.999i)8-s + (−0.560 − 1.53i)9-s + (0.291 + 0.956i)10-s + (1.06 − 0.616i)11-s + (0.934 − 1.32i)12-s + (−0.719 − 1.02i)13-s + (0.136 − 0.0365i)14-s + (0.501 − 1.54i)15-s + (0.999 + 0.00489i)16-s + (−0.587 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.237867 - 0.359277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.237867 - 0.359277i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.00173 + 1.41i)T \) |
| 5 | \( 1 + (2.13 - 0.654i)T \) |
| 19 | \( 1 + (-3.79 - 2.13i)T \) |
| good | 3 | \( 1 + (1.61 - 2.30i)T + (-1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (-0.0970 - 0.362i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.54 + 2.04i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.59 + 3.70i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (2.42 + 5.19i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-0.0474 - 0.542i)T + (-22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (0.432 + 1.18i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (6.81 + 3.93i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.60 - 5.60i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.53 + 8.70i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (6.17 + 0.539i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.622 + 1.33i)T + (-30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (5.83 - 0.510i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-3.62 - 1.32i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.14 - 2.63i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.94 - 4.17i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (7.20 + 8.58i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (8.29 + 5.80i)T + (24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (0.240 + 1.36i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.00 + 0.804i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (8.36 + 1.47i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (1.09 + 2.34i)T + (-62.3 + 74.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.25172234078547633237492229913, −10.30967488470166268776312266992, −9.585755576009660225677463025282, −8.723170919216977492740050287379, −7.40089041979175607869273884516, −5.76602228584272454814699283498, −4.88611132339934072612888706690, −3.93981145952400410469739235309, −3.09722753719677081955213954936, −0.36157077093383895583365085215,
1.39018341778970599157253259006, 4.03588516017115091196365142221, 4.97869406740595780755641592233, 6.23981540690858233940236786805, 7.01919993797778063707234610243, 7.45358996661404110372601816391, 8.563882026702835433283880452038, 9.512750046936696620831455501488, 11.08896170918140022212581742263, 11.85795567920183127788834820165
