| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 6.09·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 12.1·10-s − 17.9·11-s − 12·12-s + 17.6·13-s + 14·14-s − 18.2·15-s + 16·16-s + 6.89·17-s − 18·18-s + 26.1·19-s + 24.3·20-s + 21·21-s + 35.9·22-s + 23·23-s + 24·24-s − 87.9·25-s − 35.3·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.544·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.385·10-s − 0.492·11-s − 0.288·12-s + 0.377·13-s + 0.267·14-s − 0.314·15-s + 0.250·16-s + 0.0983·17-s − 0.235·18-s + 0.315·19-s + 0.272·20-s + 0.218·21-s + 0.348·22-s + 0.208·23-s + 0.204·24-s − 0.703·25-s − 0.266·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 6.09T + 125T^{2} \) |
| 11 | \( 1 + 17.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.89T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 30.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 412.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 423.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 218.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 85.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 917.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 474.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 465.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 49.5T + 9.12e5T^{2} \) |
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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
−9.347949849215868685154389071395, −8.467220141479913250944623912512, −7.52073412866014018059228248814, −6.66187966324040824223316294657, −5.88083068199741334113374229986, −5.08404054993470113990299717380, −3.68340091906618008441523379597, −2.45318883725585326288850506069, −1.26043986684487275120994263187, 0,
1.26043986684487275120994263187, 2.45318883725585326288850506069, 3.68340091906618008441523379597, 5.08404054993470113990299717380, 5.88083068199741334113374229986, 6.66187966324040824223316294657, 7.52073412866014018059228248814, 8.467220141479913250944623912512, 9.347949849215868685154389071395
