| L(s) = 1 | − 2·2-s + 4·4-s − 5·5-s + 23.6·7-s − 8·8-s + 10·10-s − 35.9·11-s + 72.2·13-s − 47.3·14-s + 16·16-s − 94.1·17-s − 141.·19-s − 20·20-s + 71.8·22-s + 124.·23-s + 25·25-s − 144.·26-s + 94.6·28-s + 24.9·29-s − 299.·31-s − 32·32-s + 188.·34-s − 118.·35-s + 169.·37-s + 282.·38-s + 40·40-s + 200.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.27·7-s − 0.353·8-s + 0.316·10-s − 0.984·11-s + 1.54·13-s − 0.903·14-s + 0.250·16-s − 1.34·17-s − 1.70·19-s − 0.223·20-s + 0.696·22-s + 1.13·23-s + 0.200·25-s − 1.09·26-s + 0.638·28-s + 0.159·29-s − 1.73·31-s − 0.176·32-s + 0.949·34-s − 0.571·35-s + 0.751·37-s + 1.20·38-s + 0.158·40-s + 0.762·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 23.6T + 343T^{2} \) |
| 11 | \( 1 + 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 24.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 169.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 176.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 572.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 6.46T + 2.05e5T^{2} \) |
| 61 | \( 1 - 531.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 154.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 115.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 440.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 65.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 543.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 430.T + 9.12e5T^{2} \) |
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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
−9.140886651364282747154342088180, −8.511387557806634528154673172954, −7.997817186010621902150572101002, −7.02149249613292109394067693873, −6.04639828545945712886392027646, −4.89169831392157938133148482008, −3.96212030270171707604471282583, −2.48850502711059673210506067302, −1.43249128682384408383589011167, 0,
1.43249128682384408383589011167, 2.48850502711059673210506067302, 3.96212030270171707604471282583, 4.89169831392157938133148482008, 6.04639828545945712886392027646, 7.02149249613292109394067693873, 7.997817186010621902150572101002, 8.511387557806634528154673172954, 9.140886651364282747154342088180
