| L(s) = 1 | − 4·2-s − 10.4·3-s + 16·4-s + 95.5·5-s + 41.6·6-s + 232.·7-s − 64·8-s − 134.·9-s − 382.·10-s + 345.·11-s − 166.·12-s − 506.·13-s − 928.·14-s − 993.·15-s + 256·16-s − 797.·17-s + 539.·18-s + 1.52e3·20-s − 2.41e3·21-s − 1.38e3·22-s − 1.19e3·23-s + 665.·24-s + 6.00e3·25-s + 2.02e3·26-s + 3.92e3·27-s + 3.71e3·28-s + 425.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.667·3-s + 0.5·4-s + 1.70·5-s + 0.471·6-s + 1.79·7-s − 0.353·8-s − 0.554·9-s − 1.20·10-s + 0.861·11-s − 0.333·12-s − 0.830·13-s − 1.26·14-s − 1.14·15-s + 0.250·16-s − 0.668·17-s + 0.392·18-s + 0.854·20-s − 1.19·21-s − 0.609·22-s − 0.469·23-s + 0.235·24-s + 1.92·25-s + 0.587·26-s + 1.03·27-s + 0.895·28-s + 0.0939·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.203350198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.203350198\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 10.4T + 243T^{2} \) |
| 5 | \( 1 - 95.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 232.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 345.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 506.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 797.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 425.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.64e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.60e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.90e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 9.34e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.32e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.99e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.62e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.41e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.69e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.528990685999075854232429426846, −8.986156690947470781562118167288, −8.038852558169241461504449182063, −6.97445875926586055460375533673, −6.00364639260350287618542241093, −5.41465883421031532904795520635, −4.48786026714733622864546765851, −2.44226843903405922132043332830, −1.80651712697116547078638710878, −0.827479324483089002067236983150,
0.827479324483089002067236983150, 1.80651712697116547078638710878, 2.44226843903405922132043332830, 4.48786026714733622864546765851, 5.41465883421031532904795520635, 6.00364639260350287618542241093, 6.97445875926586055460375533673, 8.038852558169241461504449182063, 8.986156690947470781562118167288, 9.528990685999075854232429426846
