| L(s) = 1 | + (1.48 − 1.01i)2-s + (1.80 − 1.67i)3-s + (0.446 − 1.13i)4-s + (−3.30 − 1.01i)5-s + (0.984 − 4.31i)6-s + (−2.12 − 1.58i)7-s + (0.311 + 1.36i)8-s + (0.229 − 3.06i)9-s + (−5.92 + 1.82i)10-s + (−0.427 − 5.70i)11-s + (−1.09 − 2.80i)12-s + (0.900 + 0.433i)13-s + (−4.74 − 0.203i)14-s + (−7.67 + 3.69i)15-s + (3.62 + 3.36i)16-s + (2.37 + 0.358i)17-s + ⋯ |
| L(s) = 1 | + (1.04 − 0.714i)2-s + (1.04 − 0.967i)3-s + (0.223 − 0.568i)4-s + (−1.47 − 0.455i)5-s + (0.401 − 1.76i)6-s + (−0.801 − 0.598i)7-s + (0.109 + 0.481i)8-s + (0.0765 − 1.02i)9-s + (−1.87 + 0.578i)10-s + (−0.128 − 1.72i)11-s + (−0.317 − 0.808i)12-s + (0.249 + 0.120i)13-s + (−1.26 − 0.0542i)14-s + (−1.98 + 0.954i)15-s + (0.907 + 0.841i)16-s + (0.576 + 0.0868i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.517234 - 2.38361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.517234 - 2.38361i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.12 + 1.58i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| good | 2 | \( 1 + (-1.48 + 1.01i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (-1.80 + 1.67i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (3.30 + 1.01i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (0.427 + 5.70i)T + (-10.8 + 1.63i)T^{2} \) |
| 17 | \( 1 + (-2.37 - 0.358i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (2.59 - 4.50i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.85 + 1.03i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.83 + 3.55i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (2.85 + 4.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.54 + 9.03i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-1.30 - 5.70i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-2.24 + 9.84i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (3.94 - 2.69i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-0.0686 + 0.175i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (-7.44 + 2.29i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (1.88 + 4.79i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-3.89 - 6.74i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.18 - 6.50i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-9.54 - 6.50i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (0.667 - 1.15i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.46 + 0.706i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.659 - 8.79i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
| 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
−10.65437119306986070528781719523, −9.036229883504078622127138891053, −8.200037684554518514379944004701, −7.85206889990663964499731570092, −6.69398619689368821834003713004, −5.46831658193804848794766074829, −3.85432263175784234496512456824, −3.65931414295902704912176027479, −2.65310872907550342784247017367, −0.885205996897175039945402287291,
2.94074532199675377937022839360, 3.47323200180505229030091106118, 4.48558481606413586245898893561, 5.04076282833814342464243994133, 6.73672255959128328442667095416, 7.15952786754751497028563242979, 8.253266114259917737856702646441, 9.200600717641867378346384244538, 9.953386575635856356151251510956, 10.85435654819409134486210737716
