| L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s − 5·11-s − 13-s − 4·14-s + 16-s + 19-s − 20-s − 5·22-s − 3·23-s + 25-s − 26-s − 4·28-s − 5·29-s + 5·31-s + 32-s + 4·35-s + 3·37-s + 38-s − 40-s − 6·41-s + 5·43-s − 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.50·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.229·19-s − 0.223·20-s − 1.06·22-s − 0.625·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s − 0.928·29-s + 0.898·31-s + 0.176·32-s + 0.676·35-s + 0.493·37-s + 0.162·38-s − 0.158·40-s − 0.937·41-s + 0.762·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−12.96242894250997, −12.69584503433837, −12.26612432180731, −11.74003508570684, −11.37603662221093, −10.67471781080644, −10.35501108504776, −9.973265338180937, −9.504374330513884, −8.971636817161715, −8.323366244593085, −7.882453583406884, −7.344112393997150, −7.114386036583232, −6.398585029442067, −5.993890795286851, −5.589979452879896, −5.056901119038800, −4.434998948666886, −4.040340386307057, −3.365813430508697, −2.944112616259890, −2.636534079728334, −1.952712597784567, −1.069402451606288, 0, 0,
1.069402451606288, 1.952712597784567, 2.636534079728334, 2.944112616259890, 3.365813430508697, 4.040340386307057, 4.434998948666886, 5.056901119038800, 5.589979452879896, 5.993890795286851, 6.398585029442067, 7.114386036583232, 7.344112393997150, 7.882453583406884, 8.323366244593085, 8.971636817161715, 9.504374330513884, 9.973265338180937, 10.35501108504776, 10.67471781080644, 11.37603662221093, 11.74003508570684, 12.26612432180731, 12.69584503433837, 12.96242894250997
