L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 5·11-s + 12-s + 16-s + 4·17-s − 18-s + 8·19-s − 5·22-s + 4·23-s − 24-s + 27-s − 5·29-s + 3·31-s − 32-s + 5·33-s − 4·34-s + 36-s + 4·37-s − 8·38-s − 2·43-s + 5·44-s − 4·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.50·11-s + 0.288·12-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.83·19-s − 1.06·22-s + 0.834·23-s − 0.204·24-s + 0.192·27-s − 0.928·29-s + 0.538·31-s − 0.176·32-s + 0.870·33-s − 0.685·34-s + 1/6·36-s + 0.657·37-s − 1.29·38-s − 0.304·43-s + 0.753·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.452519931\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.452519931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−7.81967241498152900815704220628, −7.40759340662978289434037906130, −6.74693828644844264667222732489, −5.92833840279769606874916637934, −5.18371661920317724416864995160, −4.11308576803390779259894475306, −3.39928963086299127374750424516, −2.73953911727052120787343823893, −1.50567632180377324909373109770, −0.973457831401776018640109900133,
0.973457831401776018640109900133, 1.50567632180377324909373109770, 2.73953911727052120787343823893, 3.39928963086299127374750424516, 4.11308576803390779259894475306, 5.18371661920317724416864995160, 5.92833840279769606874916637934, 6.74693828644844264667222732489, 7.40759340662978289434037906130, 7.81967241498152900815704220628