| L(s) = 1 | + (−1.30 + 0.951i)2-s + (2.95 − 1.31i)3-s + (0.190 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (−2.61 + 4.53i)6-s + (0.157 + 0.175i)7-s + (−0.690 − 2.12i)8-s + (4.99 − 5.55i)9-s + (1.47 + 0.658i)10-s + (1.95 − 0.415i)11-s + (−0.209 − 1.98i)12-s + (−0.338 + 3.21i)13-s + (−0.373 − 0.0794i)14-s + (−2.61 − 1.90i)15-s + (3.92 + 2.85i)16-s + (0.747 + 0.158i)17-s + ⋯ |
| L(s) = 1 | + (−0.925 + 0.672i)2-s + (1.70 − 0.759i)3-s + (0.0954 − 0.293i)4-s + (−0.223 − 0.387i)5-s + (−1.06 + 1.85i)6-s + (0.0597 + 0.0663i)7-s + (−0.244 − 0.751i)8-s + (1.66 − 1.85i)9-s + (0.467 + 0.208i)10-s + (0.589 − 0.125i)11-s + (−0.0603 − 0.574i)12-s + (−0.0938 + 0.892i)13-s + (−0.0998 − 0.0212i)14-s + (−0.675 − 0.491i)15-s + (0.981 + 0.713i)16-s + (0.181 + 0.0385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70496 - 0.275511i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70496 - 0.275511i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.30 - 0.951i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-2.95 + 1.31i)T + (2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.157 - 0.175i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-1.95 + 0.415i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.338 - 3.21i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-0.747 - 0.158i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.233 - 2.22i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.76 + 5.42i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.23 + 1.62i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.39 - 2.84i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.129 - 1.22i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (2 + 1.45i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-7.00 + 7.78i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 0.909i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 8.18T + 61T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.14 - 6.82i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-8.28 + 1.76i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (11.4 + 2.43i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-13.6 - 6.07i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (3.61 - 11.1i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (4.92 - 15.1i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.484813365600693533121657082539, −8.860709610671396030333403916115, −8.319886547756006381700471053969, −7.80143734952898177262368619920, −6.84453833934405824115446081525, −6.36642021827019285491018423839, −4.35652633187241642517758580828, −3.56023356987468585740590288174, −2.29607635489839945317745321525, −1.02874398411419716296613537054,
1.47557155684217227106171692713, 2.69264545887661440283173261105, 3.31145309853263925104729576062, 4.39582506513835105745450693415, 5.58083511073306263934370106308, 7.33324318063674385476533728681, 7.80263293308365736895663484499, 8.809636851381838742343708043772, 9.179168606874644148307493889177, 9.946329193947377067355622730604
