| L(s) = 1 | + 2-s + 4-s + 0.800·5-s + 4.18·7-s + 8-s + 0.800·10-s + 4.10·11-s − 1.66·13-s + 4.18·14-s + 16-s + 2.09·17-s − 3.52·19-s + 0.800·20-s + 4.10·22-s + 4.08·23-s − 4.35·25-s − 1.66·26-s + 4.18·28-s − 8.69·29-s − 8.37·31-s + 32-s + 2.09·34-s + 3.34·35-s − 5.49·37-s − 3.52·38-s + 0.800·40-s + 10.6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.357·5-s + 1.58·7-s + 0.353·8-s + 0.252·10-s + 1.23·11-s − 0.461·13-s + 1.11·14-s + 0.250·16-s + 0.507·17-s − 0.808·19-s + 0.178·20-s + 0.875·22-s + 0.852·23-s − 0.871·25-s − 0.326·26-s + 0.791·28-s − 1.61·29-s − 1.50·31-s + 0.176·32-s + 0.358·34-s + 0.566·35-s − 0.904·37-s − 0.571·38-s + 0.126·40-s + 1.67·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.460877528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.460877528\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 0.800T + 5T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 - 2.09T + 17T^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 - 4.08T + 23T^{2} \) |
| 29 | \( 1 + 8.69T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 + 5.49T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 5.66T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 4.00T + 79T^{2} \) |
| 83 | \( 1 - 1.79T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−9.393756722932582715500410195185, −8.791413307087428589666711106644, −7.65527338405879061000405359369, −7.18886744557040774692936635537, −5.96851885607966270940226069073, −5.38320433529461767612931320366, −4.41914054235808060528857932204, −3.73496608141283920020099608982, −2.22639147183055954391838170671, −1.45167455826061316949497537987,
1.45167455826061316949497537987, 2.22639147183055954391838170671, 3.73496608141283920020099608982, 4.41914054235808060528857932204, 5.38320433529461767612931320366, 5.96851885607966270940226069073, 7.18886744557040774692936635537, 7.65527338405879061000405359369, 8.791413307087428589666711106644, 9.393756722932582715500410195185
