| L(s) = 1 | − 4·2-s + 4·3-s + 16·4-s − 68·5-s − 16·6-s + 82·7-s − 64·8-s − 227·9-s + 272·10-s + 390·11-s + 64·12-s − 328·14-s − 272·15-s + 256·16-s − 1.73e3·17-s + 908·18-s + 1.07e3·19-s − 1.08e3·20-s + 328·21-s − 1.56e3·22-s − 2.10e3·23-s − 256·24-s + 1.49e3·25-s − 1.88e3·27-s + 1.31e3·28-s − 1.69e3·29-s + 1.08e3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.256·3-s + 1/2·4-s − 1.21·5-s − 0.181·6-s + 0.632·7-s − 0.353·8-s − 0.934·9-s + 0.860·10-s + 0.971·11-s + 0.128·12-s − 0.447·14-s − 0.312·15-s + 1/4·16-s − 1.45·17-s + 0.660·18-s + 0.682·19-s − 0.608·20-s + 0.162·21-s − 0.687·22-s − 0.829·23-s − 0.0907·24-s + 0.479·25-s − 0.496·27-s + 0.316·28-s − 0.373·29-s + 0.220·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9754766409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9754766409\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{5} T^{2} \) |
| 5 | \( 1 + 68 T + p^{5} T^{2} \) |
| 7 | \( 1 - 82 T + p^{5} T^{2} \) |
| 11 | \( 1 - 390 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1738 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1074 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2104 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1690 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1430 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8852 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6760 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16916 T + p^{5} T^{2} \) |
| 47 | \( 1 + 25158 T + p^{5} T^{2} \) |
| 53 | \( 1 - 38214 T + p^{5} T^{2} \) |
| 59 | \( 1 - 21286 T + p^{5} T^{2} \) |
| 61 | \( 1 + 5458 T + p^{5} T^{2} \) |
| 67 | \( 1 + 44542 T + p^{5} T^{2} \) |
| 71 | \( 1 - 17790 T + p^{5} T^{2} \) |
| 73 | \( 1 - 31064 T + p^{5} T^{2} \) |
| 79 | \( 1 + 45360 T + p^{5} T^{2} \) |
| 83 | \( 1 - 124546 T + p^{5} T^{2} \) |
| 89 | \( 1 + 18744 T + p^{5} T^{2} \) |
| 97 | \( 1 - 121488 T + p^{5} T^{2} \) |
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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
−10.95765132372656315009809653338, −9.562017649552566375919476184828, −8.672433550967950536623230641817, −8.080597551754881335109033869998, −7.18380258388319021430183366841, −6.04965797397424007708346045658, −4.52505307535669163523688403516, −3.48827405918081510715783656158, −2.10118700321670276952613171745, −0.58144755990398679628433328036,
0.58144755990398679628433328036, 2.10118700321670276952613171745, 3.48827405918081510715783656158, 4.52505307535669163523688403516, 6.04965797397424007708346045658, 7.18380258388319021430183366841, 8.080597551754881335109033869998, 8.672433550967950536623230641817, 9.562017649552566375919476184828, 10.95765132372656315009809653338
