| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 61·5-s − 36·6-s − 49·7-s + 64·8-s + 81·9-s − 244·10-s + 181·11-s − 144·12-s − 169·13-s − 196·14-s + 549·15-s + 256·16-s − 1.96e3·17-s + 324·18-s + 2.19e3·19-s − 976·20-s + 441·21-s + 724·22-s − 4.01e3·23-s − 576·24-s + 596·25-s − 676·26-s − 729·27-s − 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.09·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.771·10-s + 0.451·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.630·15-s + 1/4·16-s − 1.64·17-s + 0.235·18-s + 1.39·19-s − 0.545·20-s + 0.218·21-s + 0.318·22-s − 1.58·23-s − 0.204·24-s + 0.190·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.521175911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.521175911\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 61 T + p^{5} T^{2} \) |
| 11 | \( 1 - 181 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1965 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2193 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4015 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6841 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8992 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9753 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9900 T + p^{5} T^{2} \) |
| 43 | \( 1 - 325 p T + p^{5} T^{2} \) |
| 47 | \( 1 - 18808 T + p^{5} T^{2} \) |
| 53 | \( 1 - 2082 T + p^{5} T^{2} \) |
| 59 | \( 1 - 31700 T + p^{5} T^{2} \) |
| 61 | \( 1 - 21577 T + p^{5} T^{2} \) |
| 67 | \( 1 + 49694 T + p^{5} T^{2} \) |
| 71 | \( 1 - 53500 T + p^{5} T^{2} \) |
| 73 | \( 1 - 60137 T + p^{5} T^{2} \) |
| 79 | \( 1 - 17678 T + p^{5} T^{2} \) |
| 83 | \( 1 - 80658 T + p^{5} T^{2} \) |
| 89 | \( 1 - 81690 T + p^{5} T^{2} \) |
| 97 | \( 1 - 38054 T + p^{5} T^{2} \) |
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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
−10.22628205622163318763550904305, −9.190762236292879249792934868336, −7.976788292391071029952160201905, −7.14243002867177825830378785234, −6.34944233145685353404764664846, −5.29925051837346259857472827978, −4.23317937711280155412493491617, −3.61848399611833845399272055200, −2.16368654863231641862305599141, −0.54608968487364449544237765830,
0.54608968487364449544237765830, 2.16368654863231641862305599141, 3.61848399611833845399272055200, 4.23317937711280155412493491617, 5.29925051837346259857472827978, 6.34944233145685353404764664846, 7.14243002867177825830378785234, 7.976788292391071029952160201905, 9.190762236292879249792934868336, 10.22628205622163318763550904305
