| 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
−10.53667140852467175621436954854, −9.027689500109533684574849453825, −8.885018571560620774449739320321, −7.982794133976231156048403925403, −7.16821837841555830405106084250, −6.47305047814134919819396715917, −5.60121868748528274485580743174, −4.79905037621228732448881765091, −3.08260787004675656822097446825, −2.17209430329369904193463539426,
0.51640874768069720911669268890, 1.47008565871102980640947155680, 3.19351678662266765676176290462, 3.72114951142825520063055138245, 4.62793298723284009711347114474, 6.22577267057965768849958242724, 6.92357509166578037094905167736, 8.330199934592437742206339614347, 8.747575130360014468628294433334, 9.773993188700363206674691624920
