| L(s) = 1 | + 4.88·2-s − 3·3-s + 15.8·4-s + 5·5-s − 14.6·6-s + 5.48·7-s + 38.3·8-s + 9·9-s + 24.4·10-s + 50.1·11-s − 47.5·12-s − 20.7·13-s + 26.7·14-s − 15·15-s + 60.3·16-s − 18.8·17-s + 43.9·18-s + 78.8·19-s + 79.2·20-s − 16.4·21-s + 244.·22-s − 6.33·23-s − 114.·24-s + 25·25-s − 101.·26-s − 27·27-s + 86.9·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.577·3-s + 1.98·4-s + 0.447·5-s − 0.996·6-s + 0.296·7-s + 1.69·8-s + 0.333·9-s + 0.772·10-s + 1.37·11-s − 1.14·12-s − 0.442·13-s + 0.511·14-s − 0.258·15-s + 0.942·16-s − 0.268·17-s + 0.575·18-s + 0.951·19-s + 0.885·20-s − 0.171·21-s + 2.37·22-s − 0.0574·23-s − 0.977·24-s + 0.200·25-s − 0.763·26-s − 0.192·27-s + 0.586·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.397173343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.397173343\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 4.88T + 8T^{2} \) |
| 7 | \( 1 - 5.48T + 343T^{2} \) |
| 11 | \( 1 - 50.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 78.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.33T + 1.21e4T^{2} \) |
| 31 | \( 1 - 310.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 338.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 507.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 112.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 342.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 183.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 622.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 739.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 906.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 76.0T + 9.12e5T^{2} \) |
| 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
−11.20466149468208511437419850500, −10.11528034943950327350553961328, −9.033399839563561127542577279213, −7.43211937681196381835273539961, −6.53635187264162886694540698789, −5.85428167098803690945895068284, −4.86405504162719696594876375774, −4.10687724538625121853682746059, −2.81728917315718518384932292266, −1.39753963967844542091092476708,
1.39753963967844542091092476708, 2.81728917315718518384932292266, 4.10687724538625121853682746059, 4.86405504162719696594876375774, 5.85428167098803690945895068284, 6.53635187264162886694540698789, 7.43211937681196381835273539961, 9.033399839563561127542577279213, 10.11528034943950327350553961328, 11.20466149468208511437419850500
