| L(s) = 1 | + 9.07·5-s + 7·7-s − 36.1·11-s − 63.0·13-s + 63.0·17-s + 111.·19-s − 76.8·23-s − 42.6·25-s + 92.9·29-s − 16.0·31-s + 63.5·35-s − 88.3·37-s − 214.·41-s + 55.7·43-s + 106.·47-s + 49·49-s − 32.7·53-s − 328.·55-s + 5.36·59-s − 239.·61-s − 572.·65-s + 410.·67-s − 48.1·71-s − 908.·73-s − 253.·77-s − 580.·79-s − 730.·83-s + ⋯ |
| L(s) = 1 | + 0.811·5-s + 0.377·7-s − 0.991·11-s − 1.34·13-s + 0.899·17-s + 1.34·19-s − 0.697·23-s − 0.341·25-s + 0.595·29-s − 0.0931·31-s + 0.306·35-s − 0.392·37-s − 0.816·41-s + 0.197·43-s + 0.330·47-s + 0.142·49-s − 0.0848·53-s − 0.805·55-s + 0.0118·59-s − 0.503·61-s − 1.09·65-s + 0.748·67-s − 0.0804·71-s − 1.45·73-s − 0.374·77-s − 0.826·79-s − 0.966·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 9.07T + 125T^{2} \) |
| 11 | \( 1 + 36.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 63.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 92.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 16.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 88.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 214.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 55.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 106.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 32.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 5.36T + 2.05e5T^{2} \) |
| 61 | \( 1 + 239.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 410.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 48.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 908.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 580.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 730.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 624.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.279046471971500603600733389017, −7.63311216786724004259786136906, −6.97993543644526043697549991120, −5.69802366720823055829363112484, −5.38452838490931348423822836115, −4.49201568384350538116661922080, −3.16252826119846026223347536397, −2.37890570657122846285971167634, −1.37700653228648623308088287769, 0,
1.37700653228648623308088287769, 2.37890570657122846285971167634, 3.16252826119846026223347536397, 4.49201568384350538116661922080, 5.38452838490931348423822836115, 5.69802366720823055829363112484, 6.97993543644526043697549991120, 7.63311216786724004259786136906, 8.279046471971500603600733389017
