L(s) = 1 | + (−1.39 + 1.02i)3-s + 1.12i·5-s − 3.21·7-s + (0.886 − 2.86i)9-s − 3.06i·11-s + 0.110i·13-s + (−1.15 − 1.56i)15-s − 1.89i·17-s + (2.66 + 3.45i)19-s + (4.48 − 3.30i)21-s − 7.89i·23-s + 3.73·25-s + (1.71 + 4.90i)27-s + 1.77·29-s + 5.07i·31-s + ⋯ |
L(s) = 1 | + (−0.804 + 0.593i)3-s + 0.502i·5-s − 1.21·7-s + (0.295 − 0.955i)9-s − 0.925i·11-s + 0.0306i·13-s + (−0.298 − 0.404i)15-s − 0.460i·17-s + (0.610 + 0.791i)19-s + (0.978 − 0.721i)21-s − 1.64i·23-s + 0.747·25-s + (0.329 + 0.944i)27-s + 0.329·29-s + 0.911i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.881911 - 0.123574i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.881911 - 0.123574i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.39 - 1.02i)T \) |
| 19 | \( 1 + (-2.66 - 3.45i)T \) |
good | 5 | \( 1 - 1.12iT - 5T^{2} \) |
| 7 | \( 1 + 3.21T + 7T^{2} \) |
| 11 | \( 1 + 3.06iT - 11T^{2} \) |
| 13 | \( 1 - 0.110iT - 13T^{2} \) |
| 17 | \( 1 + 1.89iT - 17T^{2} \) |
| 23 | \( 1 + 7.89iT - 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 - 5.07iT - 31T^{2} \) |
| 37 | \( 1 + 3.59iT - 37T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 - 5.68iT - 47T^{2} \) |
| 53 | \( 1 + 0.535T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 7.02T + 61T^{2} \) |
| 67 | \( 1 + 12.5iT - 67T^{2} \) |
| 71 | \( 1 - 2.76T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 + 17.5iT - 79T^{2} \) |
| 83 | \( 1 + 8.67iT - 83T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 + 12.8iT - 97T^{2} \) |
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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
−10.26655997726920865859963625431, −9.368042327919065043682317610561, −8.608824846710127075622845131985, −7.23329994781837104208468025510, −6.43049201631957695972855848059, −5.89071809848026713382701587452, −4.78485563679905966978146814430, −3.61023890827521913035709477645, −2.88554889146180247884406554030, −0.60286678488921015899669574105,
1.01482224455774604841194560288, 2.47532752658513390089856971530, 3.88979132168654243991492023455, 5.04813088254073558526105700237, 5.78432380602556422838546960843, 6.81942441252331342906083273412, 7.28831239332384145490617484670, 8.404791673146742256872053556113, 9.545905910282530965293305093579, 9.965818232931412165079038975840