L(s) = 1 | + (0.134 + 0.232i)2-s + (−0.571 − 0.989i)3-s + (0.964 − 1.66i)4-s + 2.56·5-s + (0.153 − 0.265i)6-s + 1.05·8-s + (0.846 − 1.46i)9-s + (0.343 + 0.594i)10-s + (−1.97 − 3.41i)11-s − 2.20·12-s + (3.15 + 1.74i)13-s + (−1.46 − 2.53i)15-s + (−1.78 − 3.09i)16-s + (0.392 − 0.679i)17-s + 0.454·18-s + (−3.74 + 6.49i)19-s + ⋯ |
L(s) = 1 | + (0.0947 + 0.164i)2-s + (−0.329 − 0.571i)3-s + (0.482 − 0.834i)4-s + 1.14·5-s + (0.0625 − 0.108i)6-s + 0.372·8-s + (0.282 − 0.488i)9-s + (0.108 + 0.188i)10-s + (−0.594 − 1.03i)11-s − 0.636·12-s + (0.874 + 0.484i)13-s + (−0.378 − 0.654i)15-s + (−0.446 − 0.773i)16-s + (0.0952 − 0.164i)17-s + 0.107·18-s + (−0.859 + 1.48i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53789 - 1.15099i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53789 - 1.15099i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.15 - 1.74i)T \) |
good | 2 | \( 1 + (-0.134 - 0.232i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.571 + 0.989i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 11 | \( 1 + (1.97 + 3.41i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.392 + 0.679i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.74 - 6.49i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.97 - 6.88i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.17 + 2.03i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 + (3.37 + 5.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.21 + 2.11i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.12 + 1.94i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 1.31T + 47T^{2} \) |
| 53 | \( 1 - 9.27T + 53T^{2} \) |
| 59 | \( 1 + (4.48 - 7.76i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.72 + 8.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.676 - 1.17i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.15 - 10.6i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 0.768T + 73T^{2} \) |
| 79 | \( 1 - 6.19T + 79T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 + (-3.83 - 6.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.18 - 2.05i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.42291681610736618589527553011, −9.671054764039604353078833578090, −8.770324600513395818811817705636, −7.51489497820841464461652781794, −6.53682305896014394615375190914, −5.87825604904233719213259027710, −5.47268005349330258614069434729, −3.71908885641259085426166721123, −2.08986965524769999493118566556, −1.15551818511442488990243480011,
1.95350651225808498191316490983, 2.87624789559039379976685943527, 4.38705612538535557116335007335, 5.07704255625770446926722832577, 6.32110268338046776670878455673, 7.09363593503403342885026821609, 8.208664315642945838596089441135, 9.075396869531249836264215488442, 10.26061609411777758900255769146, 10.57050619590956214243973787064