| L(s) = 1 | + 2.31·5-s − 7-s − 1.31·11-s − 3.86·13-s + 0.326·17-s + 3.08·19-s − 3.63·23-s + 0.338·25-s − 9.50·29-s + 6.49·31-s − 2.31·35-s − 2.31·37-s + 9.49·41-s + 0.0987·43-s − 0.216·47-s + 49-s − 13.7·53-s − 3.02·55-s + 2.44·59-s − 15.3·61-s − 8.93·65-s − 5.87·67-s + 1.77·71-s − 5.99·73-s + 1.31·77-s − 14.5·79-s + 6.08·83-s + ⋯ |
| L(s) = 1 | + 1.03·5-s − 0.377·7-s − 0.395·11-s − 1.07·13-s + 0.0792·17-s + 0.707·19-s − 0.758·23-s + 0.0676·25-s − 1.76·29-s + 1.16·31-s − 0.390·35-s − 0.379·37-s + 1.48·41-s + 0.0150·43-s − 0.0316·47-s + 0.142·49-s − 1.89·53-s − 0.408·55-s + 0.318·59-s − 1.96·61-s − 1.10·65-s − 0.717·67-s + 0.210·71-s − 0.701·73-s + 0.149·77-s − 1.64·79-s + 0.667·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2.31T + 5T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 - 0.326T + 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 + 9.50T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 - 9.49T + 41T^{2} \) |
| 43 | \( 1 - 0.0987T + 43T^{2} \) |
| 47 | \( 1 + 0.216T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 5.87T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 + 5.99T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 + 5.98T + 97T^{2} \) |
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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
−7.71534126890149756709711742883, −7.43972158953188332704175078654, −6.31769677271827979130086936232, −5.84402258890045860643109574169, −5.11769833089389331967748775446, −4.29046949644474124709154607211, −3.16623431110688450534379130361, −2.42087324166564783468874527205, −1.54369255219593775387654182110, 0,
1.54369255219593775387654182110, 2.42087324166564783468874527205, 3.16623431110688450534379130361, 4.29046949644474124709154607211, 5.11769833089389331967748775446, 5.84402258890045860643109574169, 6.31769677271827979130086936232, 7.43972158953188332704175078654, 7.71534126890149756709711742883
