| L(s) = 1 | + 1.41·2-s + 1.00·4-s + 0.765·5-s + 9-s + 1.08·10-s − 1.84·11-s − 0.999·16-s − 17-s + 1.41·18-s + 0.765·20-s − 2.61·22-s − 0.765·23-s − 0.414·25-s + 1.84·29-s + 1.84·31-s − 1.41·32-s − 1.41·34-s + 1.00·36-s − 0.765·41-s − 1.84·44-s + 0.765·45-s − 1.08·46-s − 47-s + 49-s − 0.585·50-s − 1.41·55-s + 2.61·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.00·4-s + 0.765·5-s + 9-s + 1.08·10-s − 1.84·11-s − 0.999·16-s − 17-s + 1.41·18-s + 0.765·20-s − 2.61·22-s − 0.765·23-s − 0.414·25-s + 1.84·29-s + 1.84·31-s − 1.41·32-s − 1.41·34-s + 1.00·36-s − 0.765·41-s − 1.84·44-s + 0.765·45-s − 1.08·46-s − 47-s + 49-s − 0.585·50-s − 1.41·55-s + 2.61·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.995966104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995966104\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 0.765T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.84T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.765T + T^{2} \) |
| 29 | \( 1 - 1.84T + T^{2} \) |
| 31 | \( 1 - 1.84T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.765T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.84T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.28126603573702172802509605406, −10.09941779554752862531985723436, −8.731058747817948214286028450529, −7.72693161349309903227757583708, −6.60829086478723382339647557229, −5.96528377826056832212391592486, −4.89421399152344674744751737189, −4.44949415584839133562224720574, −2.99699198857455842145849223882, −2.14037838691091422830575316899,
2.14037838691091422830575316899, 2.99699198857455842145849223882, 4.44949415584839133562224720574, 4.89421399152344674744751737189, 5.96528377826056832212391592486, 6.60829086478723382339647557229, 7.72693161349309903227757583708, 8.731058747817948214286028450529, 10.09941779554752862531985723436, 10.28126603573702172802509605406
