| L(s) = 1 | + 3·3-s − 15.3·5-s + 7·7-s + 9·9-s + 17.9·11-s + 17.3·13-s − 46.1·15-s − 84.3·17-s + 159.·19-s + 21·21-s − 115.·23-s + 111.·25-s + 27·27-s − 215.·29-s + 144.·31-s + 53.9·33-s − 107.·35-s − 150.·37-s + 52.1·39-s − 229.·41-s + 411.·43-s − 138.·45-s − 45.2·47-s + 49·49-s − 253.·51-s + 604.·53-s − 277.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.37·5-s + 0.377·7-s + 0.333·9-s + 0.493·11-s + 0.371·13-s − 0.794·15-s − 1.20·17-s + 1.92·19-s + 0.218·21-s − 1.05·23-s + 0.895·25-s + 0.192·27-s − 1.37·29-s + 0.835·31-s + 0.284·33-s − 0.520·35-s − 0.670·37-s + 0.214·39-s − 0.874·41-s + 1.45·43-s − 0.458·45-s − 0.140·47-s + 0.142·49-s − 0.694·51-s + 1.56·53-s − 0.679·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.013052108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013052108\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 15.3T + 125T^{2} \) |
| 11 | \( 1 - 17.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 215.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 411.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 45.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 604.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 315.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 595.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 311.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 358.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 816.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 137.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 64.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 487.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
| 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
−9.022237748755349591832637916576, −8.446084852132257365657982708336, −7.56809655526257411901722186241, −7.18197999487862464812491685196, −5.95509789487526798952597751749, −4.76769596631267417756475781250, −3.94342666673689212587735572017, −3.33135201024269283324894278627, −1.99655004229671587863580547258, −0.69362259270608807608423706492,
0.69362259270608807608423706492, 1.99655004229671587863580547258, 3.33135201024269283324894278627, 3.94342666673689212587735572017, 4.76769596631267417756475781250, 5.95509789487526798952597751749, 7.18197999487862464812491685196, 7.56809655526257411901722186241, 8.446084852132257365657982708336, 9.022237748755349591832637916576
