| L(s) = 1 | + (1.80 + 0.146i)2-s + (−0.845 − 0.534i)3-s + (1.27 + 0.207i)4-s + (−0.0943 − 0.0836i)5-s + (−1.45 − 1.09i)6-s + (1.37 − 1.43i)7-s + (−1.24 − 0.306i)8-s + (0.428 + 0.903i)9-s + (−0.158 − 0.165i)10-s + (2.57 − 5.43i)11-s + (−0.968 − 0.858i)12-s + (3.50 − 0.853i)13-s + (2.69 − 2.38i)14-s + (0.0350 + 0.121i)15-s + (−4.66 − 1.55i)16-s + (2.59 − 2.70i)17-s + ⋯ |
| L(s) = 1 | + (1.27 + 0.103i)2-s + (−0.487 − 0.308i)3-s + (0.638 + 0.103i)4-s + (−0.0422 − 0.0373i)5-s + (−0.592 − 0.445i)6-s + (0.519 − 0.540i)7-s + (−0.439 − 0.108i)8-s + (0.142 + 0.301i)9-s + (−0.0501 − 0.0521i)10-s + (0.777 − 1.63i)11-s + (−0.279 − 0.247i)12-s + (0.971 − 0.236i)13-s + (0.720 − 0.637i)14-s + (0.00905 + 0.0312i)15-s + (−1.16 − 0.388i)16-s + (0.630 − 0.656i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.13182 - 0.978253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13182 - 0.978253i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.845 + 0.534i)T \) |
| 13 | \( 1 + (-3.50 + 0.853i)T \) |
| good | 2 | \( 1 + (-1.80 - 0.146i)T + (1.97 + 0.320i)T^{2} \) |
| 5 | \( 1 + (0.0943 + 0.0836i)T + (0.602 + 4.96i)T^{2} \) |
| 7 | \( 1 + (-1.37 + 1.43i)T + (-0.281 - 6.99i)T^{2} \) |
| 11 | \( 1 + (-2.57 + 5.43i)T + (-6.95 - 8.52i)T^{2} \) |
| 17 | \( 1 + (-2.59 + 2.70i)T + (-0.684 - 16.9i)T^{2} \) |
| 19 | \( 1 + (3.27 - 5.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.45 - 5.97i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.03 - 0.245i)T + (28.6 + 4.65i)T^{2} \) |
| 31 | \( 1 + (0.191 - 1.57i)T + (-30.0 - 7.41i)T^{2} \) |
| 37 | \( 1 + (0.832 - 0.354i)T + (25.6 - 26.6i)T^{2} \) |
| 41 | \( 1 + (3.82 + 2.41i)T + (17.5 + 37.0i)T^{2} \) |
| 43 | \( 1 + (6.41 + 2.73i)T + (29.7 + 31.0i)T^{2} \) |
| 47 | \( 1 + (3.26 - 8.60i)T + (-35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (6.23 + 1.53i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (-12.4 + 4.15i)T + (47.1 - 35.4i)T^{2} \) |
| 61 | \( 1 + (2.14 - 7.42i)T + (-51.5 - 32.6i)T^{2} \) |
| 67 | \( 1 + (-1.22 + 0.199i)T + (63.5 - 21.2i)T^{2} \) |
| 71 | \( 1 + (0.566 - 14.0i)T + (-70.7 - 5.71i)T^{2} \) |
| 73 | \( 1 + (1.79 + 2.60i)T + (-25.8 + 68.2i)T^{2} \) |
| 79 | \( 1 + (2.68 - 7.07i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-6.05 - 3.17i)T + (47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (-6.02 - 10.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.50 + 12.2i)T + (-89.2 + 38.0i)T^{2} \) |
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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
−11.25655743675184302922698088247, −10.19459407413280040763081444552, −8.814162848698843323610503477223, −8.003418573649711230979849587833, −6.67822361999316431121523218070, −5.97886924160330105921437835100, −5.22528761872612990918199962352, −4.00588852633914813662106356376, −3.26849869450288235895175622695, −1.14225161577827582214703594881,
1.95315277711802015325414893508, 3.48910246011897554517095980302, 4.55645537739119058099280548904, 5.02813884192655832763478609930, 6.27275931040556112982196284980, 6.88187528689247694751184708389, 8.500220995933672386599295590460, 9.246229117062885111412767719443, 10.40746174775724392777685802886, 11.41297502692023693793023830023
