| L(s) = 1 | − 3.48·3-s + 59.8·5-s − 145.·7-s − 230.·9-s + 121·11-s − 615.·13-s − 208.·15-s + 1.84e3·17-s + 366.·19-s + 505.·21-s + 4.51e3·23-s + 459.·25-s + 1.65e3·27-s + 1.71e3·29-s + 2.65e3·31-s − 421.·33-s − 8.68e3·35-s − 9.66e3·37-s + 2.14e3·39-s − 1.11e4·41-s + 8.36e3·43-s − 1.38e4·45-s + 2.22e3·47-s + 4.23e3·49-s − 6.41e3·51-s − 2.37e4·53-s + 7.24e3·55-s + ⋯ |
| L(s) = 1 | − 0.223·3-s + 1.07·5-s − 1.11·7-s − 0.949·9-s + 0.301·11-s − 1.01·13-s − 0.239·15-s + 1.54·17-s + 0.232·19-s + 0.250·21-s + 1.78·23-s + 0.147·25-s + 0.436·27-s + 0.379·29-s + 0.495·31-s − 0.0674·33-s − 1.19·35-s − 1.16·37-s + 0.225·39-s − 1.03·41-s + 0.690·43-s − 1.01·45-s + 0.146·47-s + 0.252·49-s − 0.345·51-s − 1.15·53-s + 0.322·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 + 3.48T + 243T^{2} \) |
| 5 | \( 1 - 59.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 145.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 615.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.84e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 366.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.51e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.11e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.36e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.95e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.39e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.83e4T + 8.58e9T^{2} \) |
| 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
−9.488935639960354611471866961435, −8.572313682538905024322007893277, −7.30819696249061388159085685193, −6.47877490986589700373909383571, −5.66527539096598159772435783057, −5.00392702833828127836653750481, −3.30268234536207269441462618458, −2.69946458312898990049857110667, −1.25630104540543583538772510331, 0,
1.25630104540543583538772510331, 2.69946458312898990049857110667, 3.30268234536207269441462618458, 5.00392702833828127836653750481, 5.66527539096598159772435783057, 6.47877490986589700373909383571, 7.30819696249061388159085685193, 8.572313682538905024322007893277, 9.488935639960354611471866961435
