| L(s) = 1 | + (1 + 2i)5-s − 3i·7-s + 3·11-s + 6i·13-s − i·17-s + 7·19-s − 8i·23-s + (−3 + 4i)25-s − 29-s − 4·31-s + (6 − 3i)35-s + 7i·37-s + 9·41-s − 4i·43-s + 3i·47-s + ⋯ |
| L(s) = 1 | + (0.447 + 0.894i)5-s − 1.13i·7-s + 0.904·11-s + 1.66i·13-s − 0.242i·17-s + 1.60·19-s − 1.66i·23-s + (−0.600 + 0.800i)25-s − 0.185·29-s − 0.718·31-s + (1.01 − 0.507i)35-s + 1.15i·37-s + 1.40·41-s − 0.609i·43-s + 0.437i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.232340953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.232340953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.030190634323177521994056444904, −7.79381064373629691518494197359, −7.06850148904837871281589222110, −6.70418243522148918370456885647, −5.94752631257776329257979988743, −4.72553405300048899068224475454, −4.03762427998746288446853130684, −3.22466986924030648047240606086, −2.10757592858891139084635845783, −1.04604581507608697921528056780,
0.880760557599741207154290674108, 1.87588625849602280214769046866, 3.02193101153615216526020339387, 3.84241719484363781823535948672, 5.12534965063954377154707748427, 5.58626587398920141093482867447, 6.01052912532045277840973151936, 7.34765532044979722471767244381, 7.947440577477874407048201272273, 8.768017286403247775874230621463
