| L(s) = 1 | + 5.23·2-s + 3·3-s + 19.4·4-s − 9.35·5-s + 15.7·6-s + 10.1·7-s + 59.7·8-s + 9·9-s − 48.9·10-s + 11·11-s + 58.2·12-s − 26.6·13-s + 53.2·14-s − 28.0·15-s + 157.·16-s − 0.411·17-s + 47.1·18-s + 34.2·19-s − 181.·20-s + 30.4·21-s + 57.5·22-s + 181.·23-s + 179.·24-s − 37.5·25-s − 139.·26-s + 27·27-s + 197.·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.577·3-s + 2.42·4-s − 0.836·5-s + 1.06·6-s + 0.548·7-s + 2.64·8-s + 0.333·9-s − 1.54·10-s + 0.301·11-s + 1.40·12-s − 0.568·13-s + 1.01·14-s − 0.482·15-s + 2.46·16-s − 0.00587·17-s + 0.617·18-s + 0.413·19-s − 2.02·20-s + 0.316·21-s + 0.558·22-s + 1.64·23-s + 1.52·24-s − 0.300·25-s − 1.05·26-s + 0.192·27-s + 1.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.701645042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.701645042\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 5.23T + 8T^{2} \) |
| 5 | \( 1 + 9.35T + 125T^{2} \) |
| 7 | \( 1 - 10.1T + 343T^{2} \) |
| 13 | \( 1 + 26.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.411T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 94.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 293.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 547.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 182.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 164.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 823.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 421.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 310.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 463.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 624.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.81e3T + 9.12e5T^{2} \) |
| 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
−8.594836480974302931812962756000, −7.49361787496838822388328398802, −7.35415621678461937954629019451, −6.26352522987925189105791813570, −5.33548431152710482913363389913, −4.52783672835070272070592829944, −4.03388278553180834854316893282, −3.07989565272948704127146938245, −2.41026324329794278494692252558, −1.12535472609149970850469583940,
1.12535472609149970850469583940, 2.41026324329794278494692252558, 3.07989565272948704127146938245, 4.03388278553180834854316893282, 4.52783672835070272070592829944, 5.33548431152710482913363389913, 6.26352522987925189105791813570, 7.35415621678461937954629019451, 7.49361787496838822388328398802, 8.594836480974302931812962756000
