| 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 | \( 1 + 8.45T + 97T^{2} \) |
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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
−10.06762546460801908986721568520, −9.654979425143925990649499075496, −8.983487196020990654764469154132, −8.075544932983619640048978918024, −7.53194967897482966988010598793, −6.33484063413714852984178950057, −5.08413098996205389642311726523, −3.97128936562935943965619530167, −2.48235705061886122077962988703, −1.43690390191908785903412591246,
0.74681932142165338360280557659, 3.10680602101191644334261162373, 3.61078858923164777550073236374, 4.68571612295322521939294330438, 6.33018861608070271230065870694, 7.08508313927859833198899943445, 8.341601006729489839153222171879, 8.841031204057257290866492825510, 9.398062450995000537296071196951, 10.36358756655106073901464697412
