| L(s) = 1 | + (0.148 + 0.107i)2-s + (−1.27 − 3.08i)3-s + (−0.607 − 1.87i)4-s + (2.05 + 0.892i)5-s + (0.142 − 0.595i)6-s + (0.968 − 1.58i)7-s + (0.224 − 0.692i)8-s + (−5.75 + 5.75i)9-s + (0.208 + 0.353i)10-s + (−1.85 + 2.17i)11-s + (−4.99 + 4.26i)12-s + (0.525 − 2.18i)13-s + (0.314 − 0.130i)14-s + (0.132 − 7.46i)15-s + (−3.07 + 2.23i)16-s + (0.954 + 0.814i)17-s + ⋯ |
| L(s) = 1 | + (0.104 + 0.0762i)2-s + (−0.737 − 1.78i)3-s + (−0.303 − 0.935i)4-s + (0.916 + 0.399i)5-s + (0.0583 − 0.243i)6-s + (0.366 − 0.597i)7-s + (0.0794 − 0.244i)8-s + (−1.91 + 1.91i)9-s + (0.0658 + 0.111i)10-s + (−0.559 + 0.654i)11-s + (−1.44 + 1.23i)12-s + (0.145 − 0.607i)13-s + (0.0839 − 0.0347i)14-s + (0.0342 − 1.92i)15-s + (−0.768 + 0.558i)16-s + (0.231 + 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.377493 - 0.946225i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.377493 - 0.946225i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.05 - 0.892i)T \) |
| 41 | \( 1 + (2.47 + 5.90i)T \) |
| good | 2 | \( 1 + (-0.148 - 0.107i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (1.27 + 3.08i)T + (-2.12 + 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.968 + 1.58i)T + (-3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (1.85 - 2.17i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.525 + 2.18i)T + (-11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (-0.954 - 0.814i)T + (2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-3.95 + 6.44i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-0.431 - 2.72i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.165 - 2.10i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (2.78 + 0.903i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.468 + 0.918i)T + (-21.7 - 29.9i)T^{2} \) |
| 43 | \( 1 + (-5.78 + 7.96i)T + (-13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-9.41 + 5.76i)T + (21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (-0.796 - 0.932i)T + (-8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (2.83 - 3.90i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.454 - 2.86i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (6.90 + 0.543i)T + (66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-6.09 - 5.20i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + 3.85iT - 73T^{2} \) |
| 79 | \( 1 + (3.40 - 8.22i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-7.14 + 7.14i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.62 - 0.869i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (1.01 - 12.8i)T + (-95.8 - 15.1i)T^{2} \) |
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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
−12.21135246820299332699166545248, −10.94123539170776690070021361687, −10.48648202779801339164347529820, −9.067057457068592110329768970925, −7.45858779356218589264089678626, −6.95839153037334864940473359694, −5.71694436145577855799477268362, −5.20027804901037639079383469214, −2.29787306765703462528656950321, −1.00695732890704997499369212486,
3.02763927403069541351177976322, 4.30641799718904577249281787787, 5.26866730116112641145455817458, 6.03262823175155371564871586707, 8.186466046360294481685168869236, 9.093985069158504181686384989335, 9.767458436681957376786606466987, 10.83579321243025571577908387185, 11.73619173992830232252336059273, 12.50059873215881085197837347562
