| L(s) = 1 | − 10·2-s − 73·3-s − 28·4-s + 295·5-s + 730·6-s − 1.37e3·7-s + 1.56e3·8-s + 3.14e3·9-s − 2.95e3·10-s + 7.64e3·11-s + 2.04e3·12-s + 1.37e4·14-s − 2.15e4·15-s − 1.20e4·16-s − 4.14e3·17-s − 3.14e4·18-s + 3.18e3·19-s − 8.26e3·20-s + 1.00e5·21-s − 7.64e4·22-s − 1.77e4·23-s − 1.13e5·24-s + 8.90e3·25-s − 6.97e4·27-s + 3.84e4·28-s − 9.33e4·29-s + 2.15e5·30-s + ⋯ |
| L(s) = 1 | − 0.883·2-s − 1.56·3-s − 0.218·4-s + 1.05·5-s + 1.37·6-s − 1.51·7-s + 1.07·8-s + 1.43·9-s − 0.932·10-s + 1.73·11-s + 0.341·12-s + 1.33·14-s − 1.64·15-s − 0.733·16-s − 0.204·17-s − 1.26·18-s + 0.106·19-s − 0.230·20-s + 2.36·21-s − 1.53·22-s − 0.304·23-s − 1.68·24-s + 0.113·25-s − 0.681·27-s + 0.330·28-s − 0.710·29-s + 1.45·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.4981903282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4981903282\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 5 p T + p^{7} T^{2} \) |
| 3 | \( 1 + 73 T + p^{7} T^{2} \) |
| 5 | \( 1 - 59 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1373 T + p^{7} T^{2} \) |
| 11 | \( 1 - 7646 T + p^{7} T^{2} \) |
| 17 | \( 1 + 4147 T + p^{7} T^{2} \) |
| 19 | \( 1 - 3186 T + p^{7} T^{2} \) |
| 23 | \( 1 + 17784 T + p^{7} T^{2} \) |
| 29 | \( 1 + 3218 p T + p^{7} T^{2} \) |
| 31 | \( 1 - 124484 T + p^{7} T^{2} \) |
| 37 | \( 1 + 273661 T + p^{7} T^{2} \) |
| 41 | \( 1 + 585816 T + p^{7} T^{2} \) |
| 43 | \( 1 + 533559 T + p^{7} T^{2} \) |
| 47 | \( 1 - 530055 T + p^{7} T^{2} \) |
| 53 | \( 1 + 615288 T + p^{7} T^{2} \) |
| 59 | \( 1 - 392514 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1878064 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3971438 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3746601 T + p^{7} T^{2} \) |
| 73 | \( 1 + 2485802 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1264456 T + p^{7} T^{2} \) |
| 83 | \( 1 + 434308 T + p^{7} T^{2} \) |
| 89 | \( 1 + 5830810 T + p^{7} T^{2} \) |
| 97 | \( 1 - 2045330 T + p^{7} T^{2} \) |
| 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
−11.32634977934019012287734902163, −10.06631185044059663623268044025, −9.786796507385403721642985359470, −8.826311051016275169757896891142, −6.85016344746831906192515124917, −6.38103008124796318073534965590, −5.31120937166109105311736925027, −3.87028102741546258170179501089, −1.60880731852827963118548813285, −0.50912080261104158854276912554,
0.50912080261104158854276912554, 1.60880731852827963118548813285, 3.87028102741546258170179501089, 5.31120937166109105311736925027, 6.38103008124796318073534965590, 6.85016344746831906192515124917, 8.826311051016275169757896891142, 9.786796507385403721642985359470, 10.06631185044059663623268044025, 11.32634977934019012287734902163
