| L(s) = 1 | − 4·2-s + 19.9·3-s + 16·4-s − 25·5-s − 79.8·6-s − 169.·7-s − 64·8-s + 155.·9-s + 100·10-s + 371.·11-s + 319.·12-s + 541.·13-s + 678.·14-s − 499.·15-s + 256·16-s − 26.6·17-s − 623.·18-s − 410.·19-s − 400·20-s − 3.38e3·21-s − 1.48e3·22-s − 529·23-s − 1.27e3·24-s + 625·25-s − 2.16e3·26-s − 1.74e3·27-s − 2.71e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.28·3-s + 0.5·4-s − 0.447·5-s − 0.905·6-s − 1.30·7-s − 0.353·8-s + 0.641·9-s + 0.316·10-s + 0.925·11-s + 0.640·12-s + 0.888·13-s + 0.925·14-s − 0.572·15-s + 0.250·16-s − 0.0223·17-s − 0.453·18-s − 0.260·19-s − 0.223·20-s − 1.67·21-s − 0.654·22-s − 0.208·23-s − 0.452·24-s + 0.200·25-s − 0.628·26-s − 0.459·27-s − 0.654·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 + 25T \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 - 19.9T + 243T^{2} \) |
| 7 | \( 1 + 169.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 371.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 541.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 26.6T + 1.41e6T^{2} \) |
| 19 | \( 1 + 410.T + 2.47e6T^{2} \) |
| 29 | \( 1 + 6.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 124.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.24e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 368.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.44e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.50e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.89e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.83e4T + 8.58e9T^{2} \) |
| 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
−10.64579251252764085697394731492, −9.407032442451427550477939637839, −9.072359548340931631075158935803, −8.074485918229511028881191444549, −7.05970405197129630782624399909, −6.06494053491146712484074291525, −3.81374994414564632308215860838, −3.20437175340088142005339780921, −1.72659722526996983920231035493, 0,
1.72659722526996983920231035493, 3.20437175340088142005339780921, 3.81374994414564632308215860838, 6.06494053491146712484074291525, 7.05970405197129630782624399909, 8.074485918229511028881191444549, 9.072359548340931631075158935803, 9.407032442451427550477939637839, 10.64579251252764085697394731492
