| L(s) = 1 | + (−0.571 − 1.75i)2-s + (0.334 + 0.371i)3-s + (−1.15 + 0.836i)4-s + (−0.603 − 1.04i)5-s + (0.461 − 0.800i)6-s + (−0.390 − 3.71i)7-s + (−0.863 − 0.627i)8-s + (0.287 − 2.73i)9-s + (−1.49 + 1.66i)10-s + (−1.69 − 0.755i)11-s + (−0.695 − 0.147i)12-s + (5.07 − 1.07i)13-s + (−6.30 + 2.80i)14-s + (0.186 − 0.573i)15-s + (−1.48 + 4.58i)16-s + (−5.17 + 2.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.404 − 1.24i)2-s + (0.192 + 0.214i)3-s + (−0.575 + 0.418i)4-s + (−0.269 − 0.467i)5-s + (0.188 − 0.326i)6-s + (−0.147 − 1.40i)7-s + (−0.305 − 0.221i)8-s + (0.0958 − 0.911i)9-s + (−0.472 + 0.524i)10-s + (−0.511 − 0.227i)11-s + (−0.200 − 0.0426i)12-s + (1.40 − 0.298i)13-s + (−1.68 + 0.750i)14-s + (0.0481 − 0.148i)15-s + (−0.372 + 1.14i)16-s + (−1.25 + 0.559i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.355638 + 0.899504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.355638 + 0.899504i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (0.571 + 1.75i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.334 - 0.371i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (0.603 + 1.04i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.390 + 3.71i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (1.69 + 0.755i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-5.07 + 1.07i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (5.17 - 2.30i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.40 - 0.299i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.86 - 2.08i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.424 - 1.30i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.25 - 3.89i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.15 - 3.50i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (6.41 + 1.36i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (1.30 - 4.02i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.35 + 12.8i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (1.45 + 1.62i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 + (1.44 + 2.49i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.940 - 8.95i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (3.84 + 1.70i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-10.2 + 4.56i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (7.01 - 7.78i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-2.18 + 1.58i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.70 + 4.87i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.773993188700363206674691624920, −8.747575130360014468628294433334, −8.330199934592437742206339614347, −6.92357509166578037094905167736, −6.22577267057965768849958242724, −4.62793298723284009711347114474, −3.72114951142825520063055138245, −3.19351678662266765676176290462, −1.47008565871102980640947155680, −0.51640874768069720911669268890,
2.17209430329369904193463539426, 3.08260787004675656822097446825, 4.79905037621228732448881765091, 5.60121868748528274485580743174, 6.47305047814134919819396715917, 7.16821837841555830405106084250, 7.982794133976231156048403925403, 8.885018571560620774449739320321, 9.027689500109533684574849453825, 10.53667140852467175621436954854
