L(s) = 1 | + (0.707 − 0.707i)3-s + (−0.162 − 2.23i)5-s − 2.93·7-s − 1.00i·9-s + (−0.663 + 0.663i)11-s + (−1.12 + 1.12i)13-s + (−1.69 − 1.46i)15-s + 7.47i·17-s + (0.423 + 0.423i)19-s + (−2.07 + 2.07i)21-s − 6.17·23-s + (−4.94 + 0.722i)25-s + (−0.707 − 0.707i)27-s + (2.95 + 2.95i)29-s − 1.82·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.0724 − 0.997i)5-s − 1.10·7-s − 0.333i·9-s + (−0.200 + 0.200i)11-s + (−0.312 + 0.312i)13-s + (−0.436 − 0.377i)15-s + 1.81i·17-s + (0.0971 + 0.0971i)19-s + (−0.453 + 0.453i)21-s − 1.28·23-s + (−0.989 + 0.144i)25-s + (−0.136 − 0.136i)27-s + (0.548 + 0.548i)29-s − 0.327·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5905642440\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5905642440\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.162 + 2.23i)T \) |
good | 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 + (0.663 - 0.663i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.12 - 1.12i)T - 13iT^{2} \) |
| 17 | \( 1 - 7.47iT - 17T^{2} \) |
| 19 | \( 1 + (-0.423 - 0.423i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 + (-2.95 - 2.95i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 + (-5.53 - 5.53i)T + 37iT^{2} \) |
| 41 | \( 1 + 12.3iT - 41T^{2} \) |
| 43 | \( 1 + (-0.897 - 0.897i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.12iT - 47T^{2} \) |
| 53 | \( 1 + (0.146 + 0.146i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.72 - 7.72i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.37 - 7.37i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.68 + 8.68i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.95iT - 71T^{2} \) |
| 73 | \( 1 - 0.174T + 73T^{2} \) |
| 79 | \( 1 + 3.06T + 79T^{2} \) |
| 83 | \( 1 + (9.18 - 9.18i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.71iT - 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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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
−9.398375954257874931205169272183, −8.514811264352726300114070269153, −8.046818336953174922918942293012, −7.06392680087289416120831894629, −6.21528248778880835988715252895, −5.55860294200663866716653326074, −4.28934033179152424698972212310, −3.68326942320699732521284996904, −2.43236636261316105042046535803, −1.36660315613552945742265410657,
0.19866245401185411132662971564, 2.41769402467427469524394225926, 2.99068494988378935448002589591, 3.79019403271799730578020654686, 4.86925532852961466818465834634, 5.94152296940027514832131413283, 6.63499072307560255874318796668, 7.47415142389748935709237080260, 8.084695897667316385751128370315, 9.294402063876572740306496148707