| L(s) = 1 | + (−0.985 + 0.148i)2-s + (1.55 + 1.06i)3-s + (−0.962 + 0.297i)4-s + (−0.0612 + 0.817i)5-s + (−1.69 − 0.814i)6-s + (−1.98 − 1.75i)7-s + (2.69 − 1.29i)8-s + (0.202 + 0.515i)9-s + (−0.0610 − 0.814i)10-s + (2.19 − 5.60i)11-s + (−1.81 − 0.559i)12-s + (0.623 + 0.781i)13-s + (2.21 + 1.43i)14-s + (−0.964 + 1.20i)15-s + (−0.800 + 0.545i)16-s + (−1.16 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.104i)2-s + (0.899 + 0.613i)3-s + (−0.481 + 0.148i)4-s + (−0.0274 + 0.365i)5-s + (−0.690 − 0.332i)6-s + (−0.749 − 0.662i)7-s + (0.954 − 0.459i)8-s + (0.0674 + 0.171i)9-s + (−0.0193 − 0.257i)10-s + (0.662 − 1.68i)11-s + (−0.524 − 0.161i)12-s + (0.172 + 0.216i)13-s + (0.591 + 0.382i)14-s + (−0.248 + 0.312i)15-s + (−0.200 + 0.136i)16-s + (−0.282 + 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04120 - 0.222789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04120 - 0.222789i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.98 + 1.75i)T \) |
| 13 | \( 1 + (-0.623 - 0.781i)T \) |
| good | 2 | \( 1 + (0.985 - 0.148i)T + (1.91 - 0.589i)T^{2} \) |
| 3 | \( 1 + (-1.55 - 1.06i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (0.0612 - 0.817i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (-2.19 + 5.60i)T + (-8.06 - 7.48i)T^{2} \) |
| 17 | \( 1 + (1.16 - 1.08i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.07 + 5.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.21 + 2.98i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.715 - 3.13i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-4.54 - 7.87i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0339 + 0.0104i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-8.05 + 3.88i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (9.72 + 4.68i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (4.38 - 0.661i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-8.48 + 2.61i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.356 - 4.76i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (8.43 + 2.60i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (5.69 + 9.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.207 - 0.907i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-9.52 - 1.43i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-2.86 + 4.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.32 + 9.18i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.44 - 3.67i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 8.45T + 97T^{2} \) |
| 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
−10.36358756655106073901464697412, −9.398062450995000537296071196951, −8.841031204057257290866492825510, −8.341601006729489839153222171879, −7.08508313927859833198899943445, −6.33018861608070271230065870694, −4.68571612295322521939294330438, −3.61078858923164777550073236374, −3.10680602101191644334261162373, −0.74681932142165338360280557659,
1.43690390191908785903412591246, 2.48235705061886122077962988703, 3.97128936562935943965619530167, 5.08413098996205389642311726523, 6.33484063413714852984178950057, 7.53194967897482966988010598793, 8.075544932983619640048978918024, 8.983487196020990654764469154132, 9.654979425143925990649499075496, 10.06762546460801908986721568520
