| L(s) = 1 | − 2·2-s − 6.50·3-s + 4·4-s + 5·5-s + 13.0·6-s + 2.73·7-s − 8·8-s + 15.2·9-s − 10·10-s − 58.0·11-s − 26.0·12-s + 60.8·13-s − 5.46·14-s − 32.5·15-s + 16·16-s − 25.5·17-s − 30.5·18-s − 135.·19-s + 20·20-s − 17.7·21-s + 116.·22-s + 23·23-s + 52.0·24-s + 25·25-s − 121.·26-s + 76.2·27-s + 10.9·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.25·3-s + 0.5·4-s + 0.447·5-s + 0.884·6-s + 0.147·7-s − 0.353·8-s + 0.565·9-s − 0.316·10-s − 1.59·11-s − 0.625·12-s + 1.29·13-s − 0.104·14-s − 0.559·15-s + 0.250·16-s − 0.364·17-s − 0.400·18-s − 1.63·19-s + 0.223·20-s − 0.184·21-s + 1.12·22-s + 0.208·23-s + 0.442·24-s + 0.200·25-s − 0.917·26-s + 0.543·27-s + 0.0737·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7068210779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7068210779\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 6.50T + 27T^{2} \) |
| 7 | \( 1 - 2.73T + 343T^{2} \) |
| 11 | \( 1 + 58.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 25.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 135.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 76.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 411.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 279.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 444.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 60.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 417.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 474.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 430.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 835.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 169.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 623.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.67e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−11.30022138604171927542297853441, −10.85256829840449465816549390735, −10.14371333386047858143649086692, −8.781708415186736967016304768745, −7.87477636028186694918466906435, −6.41428386120771470466303122723, −5.86931720952461481216839434665, −4.60101127242223195463934339678, −2.48331950479038644699031082063, −0.72091258386962182676758978260,
0.72091258386962182676758978260, 2.48331950479038644699031082063, 4.60101127242223195463934339678, 5.86931720952461481216839434665, 6.41428386120771470466303122723, 7.87477636028186694918466906435, 8.781708415186736967016304768745, 10.14371333386047858143649086692, 10.85256829840449465816549390735, 11.30022138604171927542297853441
