| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 25·5-s − 36·6-s − 64·8-s − 162·9-s + 100·10-s − 187·11-s + 144·12-s − 627·13-s − 225·15-s + 256·16-s − 1.81e3·17-s + 648·18-s − 258·19-s − 400·20-s + 748·22-s + 2.97e3·23-s − 576·24-s + 625·25-s + 2.50e3·26-s − 3.64e3·27-s + 1.29e3·29-s + 900·30-s − 1.91e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.465·11-s + 0.288·12-s − 1.02·13-s − 0.258·15-s + 1/4·16-s − 1.52·17-s + 0.471·18-s − 0.163·19-s − 0.223·20-s + 0.329·22-s + 1.17·23-s − 0.204·24-s + 1/5·25-s + 0.727·26-s − 0.962·27-s + 0.286·29-s + 0.182·30-s − 0.358·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9233279383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9233279383\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 11 | \( 1 + 17 p T + p^{5} T^{2} \) |
| 13 | \( 1 + 627 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1813 T + p^{5} T^{2} \) |
| 19 | \( 1 + 258 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2970 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1299 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1916 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6578 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6676 T + p^{5} T^{2} \) |
| 43 | \( 1 - 3178 T + p^{5} T^{2} \) |
| 47 | \( 1 - 22001 T + p^{5} T^{2} \) |
| 53 | \( 1 - 26168 T + p^{5} T^{2} \) |
| 59 | \( 1 + 3932 T + p^{5} T^{2} \) |
| 61 | \( 1 - 48740 T + p^{5} T^{2} \) |
| 67 | \( 1 + 44832 T + p^{5} T^{2} \) |
| 71 | \( 1 - 63736 T + p^{5} T^{2} \) |
| 73 | \( 1 + 60470 T + p^{5} T^{2} \) |
| 79 | \( 1 + 43721 T + p^{5} T^{2} \) |
| 83 | \( 1 + 1172 p T + p^{5} T^{2} \) |
| 89 | \( 1 + 45560 T + p^{5} T^{2} \) |
| 97 | \( 1 - 57295 T + p^{5} T^{2} \) |
| 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
−10.06946045990873309741306223506, −9.008714711326159717011128606890, −8.576298291749840449319879406820, −7.55116738608735753244862317396, −6.85931802459737413477618739223, −5.52708750154168906910146212207, −4.33216611808213363172542923865, −2.92884824972011609889419047376, −2.21876074194027474424678774896, −0.49528892512479640959912457415,
0.49528892512479640959912457415, 2.21876074194027474424678774896, 2.92884824972011609889419047376, 4.33216611808213363172542923865, 5.52708750154168906910146212207, 6.85931802459737413477618739223, 7.55116738608735753244862317396, 8.576298291749840449319879406820, 9.008714711326159717011128606890, 10.06946045990873309741306223506
