| L(s) = 1 | + (0.568 − 0.822i)2-s + (0.623 − 0.153i)3-s + (−0.354 − 0.935i)4-s + (0.662 + 0.586i)5-s + (0.227 − 0.600i)6-s + (2.87 − 0.707i)7-s + (−0.970 − 0.239i)8-s + (−2.29 + 1.20i)9-s + (0.859 − 0.211i)10-s + (−0.0563 + 0.0499i)11-s + (−0.364 − 0.528i)12-s + (−2.00 − 5.28i)13-s + (1.04 − 2.76i)14-s + (0.502 + 0.263i)15-s + (−0.748 + 0.663i)16-s + (1.52 + 4.02i)17-s + ⋯ |
| L(s) = 1 | + (0.401 − 0.581i)2-s + (0.359 − 0.0886i)3-s + (−0.177 − 0.467i)4-s + (0.296 + 0.262i)5-s + (0.0929 − 0.245i)6-s + (1.08 − 0.267i)7-s + (−0.343 − 0.0846i)8-s + (−0.763 + 0.400i)9-s + (0.271 − 0.0669i)10-s + (−0.0169 + 0.0150i)11-s + (−0.105 − 0.152i)12-s + (−0.556 − 1.46i)13-s + (0.280 − 0.739i)14-s + (0.129 + 0.0681i)15-s + (−0.187 + 0.165i)16-s + (0.370 + 0.975i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41055 - 0.649358i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41055 - 0.649358i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.568 + 0.822i)T \) |
| 79 | \( 1 + (-2.73 + 8.45i)T \) |
| good | 3 | \( 1 + (-0.623 + 0.153i)T + (2.65 - 1.39i)T^{2} \) |
| 5 | \( 1 + (-0.662 - 0.586i)T + (0.602 + 4.96i)T^{2} \) |
| 7 | \( 1 + (-2.87 + 0.707i)T + (6.19 - 3.25i)T^{2} \) |
| 11 | \( 1 + (0.0563 - 0.0499i)T + (1.32 - 10.9i)T^{2} \) |
| 13 | \( 1 + (2.00 + 5.28i)T + (-9.73 + 8.62i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 4.02i)T + (-12.7 + 11.2i)T^{2} \) |
| 19 | \( 1 + (0.699 - 5.75i)T + (-18.4 - 4.54i)T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 + (-9.36 - 4.91i)T + (16.4 + 23.8i)T^{2} \) |
| 31 | \( 1 + (3.33 - 4.82i)T + (-10.9 - 28.9i)T^{2} \) |
| 37 | \( 1 + (-0.0946 + 0.779i)T + (-35.9 - 8.85i)T^{2} \) |
| 41 | \( 1 + (1.76 + 1.56i)T + (4.94 + 40.7i)T^{2} \) |
| 43 | \( 1 + (4.14 + 3.67i)T + (5.18 + 42.6i)T^{2} \) |
| 47 | \( 1 + (0.00220 + 0.0181i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-3.70 - 0.913i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (-1.83 + 4.84i)T + (-44.1 - 39.1i)T^{2} \) |
| 61 | \( 1 + (-0.251 + 2.07i)T + (-59.2 - 14.5i)T^{2} \) |
| 67 | \( 1 + (2.39 + 3.46i)T + (-23.7 + 62.6i)T^{2} \) |
| 71 | \( 1 + (1.28 + 0.315i)T + (62.8 + 32.9i)T^{2} \) |
| 73 | \( 1 + (-5.51 + 14.5i)T + (-54.6 - 48.4i)T^{2} \) |
| 83 | \( 1 + (-3.68 - 9.70i)T + (-62.1 + 55.0i)T^{2} \) |
| 89 | \( 1 + (-0.134 + 0.0332i)T + (78.8 - 41.3i)T^{2} \) |
| 97 | \( 1 + (-2.19 + 18.0i)T + (-94.1 - 23.2i)T^{2} \) |
| 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
−12.64963093857973444577341771484, −11.93487694070870412216918817257, −10.47869643848459287814007043976, −10.33990906026195839324302011030, −8.434360099225977843665073980755, −7.88059023312376886416635334228, −6.01373621641194949009321793097, −4.99391922690055363298588790857, −3.41035355202008181845512870340, −1.95692318320326561768009779796,
2.44500058299796295211184965001, 4.32353341238910297028440391970, 5.32293303484547358148511105425, 6.62401058493281120110077920104, 7.88026776692006112332745778148, 8.845121740290671052760573569137, 9.660477904479692960877803899148, 11.60757494853289657787197562439, 11.78367030248372199458913000219, 13.43331714526023270345378631023
