| L(s) = 1 | − 4·2-s + 29.8·3-s + 16·4-s + 21·5-s − 119.·6-s − 64·8-s + 645.·9-s − 84·10-s + 331.·11-s + 476.·12-s − 66.8·13-s + 625.·15-s + 256·16-s − 240.·17-s − 2.58e3·18-s − 441.·19-s + 336·20-s − 1.32e3·22-s − 1.07e3·23-s − 1.90e3·24-s − 2.68e3·25-s + 267.·26-s + 1.19e4·27-s + 1.79e3·29-s − 2.50e3·30-s + 5.68e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.91·3-s + 0.5·4-s + 0.375·5-s − 1.35·6-s − 0.353·8-s + 2.65·9-s − 0.265·10-s + 0.826·11-s + 0.955·12-s − 0.109·13-s + 0.718·15-s + 0.250·16-s − 0.201·17-s − 1.87·18-s − 0.280·19-s + 0.187·20-s − 0.584·22-s − 0.422·23-s − 0.675·24-s − 0.858·25-s + 0.0775·26-s + 3.16·27-s + 0.395·29-s − 0.507·30-s + 1.06·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.857224195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.857224195\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 29.8T + 243T^{2} \) |
| 5 | \( 1 - 21T + 3.12e3T^{2} \) |
| 11 | \( 1 - 331.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 66.8T + 3.71e5T^{2} \) |
| 17 | \( 1 + 240.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 441.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.20e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.29e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.18e4T + 8.58e9T^{2} \) |
| 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
−13.28273545224890244975325750588, −11.95950668698245076315898920684, −10.24786076388569582905627734501, −9.486434220274517153296047473151, −8.637272735487623794522720967327, −7.74720702211958918746642925833, −6.53791483958374596983253057837, −4.12623272533962640301630900794, −2.71593181875707664145104247036, −1.52930606058188947772642361523,
1.52930606058188947772642361523, 2.71593181875707664145104247036, 4.12623272533962640301630900794, 6.53791483958374596983253057837, 7.74720702211958918746642925833, 8.637272735487623794522720967327, 9.486434220274517153296047473151, 10.24786076388569582905627734501, 11.95950668698245076315898920684, 13.28273545224890244975325750588
