| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 2.86·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s + 5.72·10-s + 59.1·11-s + 12·12-s − 13·13-s − 14·14-s − 8.59·15-s + 16·16-s − 32.4·17-s − 18·18-s + 63.0·19-s − 11.4·20-s + 21·21-s − 118.·22-s − 26.4·23-s − 24·24-s − 116.·25-s + 26·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.256·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.181·10-s + 1.61·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.147·15-s + 0.250·16-s − 0.463·17-s − 0.235·18-s + 0.761·19-s − 0.128·20-s + 0.218·21-s − 1.14·22-s − 0.239·23-s − 0.204·24-s − 0.934·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.927152831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927152831\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 2.86T + 125T^{2} \) |
| 11 | \( 1 - 59.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 32.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 63.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 86.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 78.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 450.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.48T + 7.95e4T^{2} \) |
| 47 | \( 1 + 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 496.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 204.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 356.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 50.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.16e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 871.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 268.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 396.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 58.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 387.T + 9.12e5T^{2} \) |
| 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
−10.14416590881543937337458946076, −9.394295258830305089581961707551, −8.710062309449484082939250070180, −7.82691177802607610854773959211, −7.01598564993259916972914767975, −6.05120035480725771543282038318, −4.52625526011822258667280086553, −3.52192649435449321981274035567, −2.14928711954148034554310854156, −0.960679498143732420020062514663,
0.960679498143732420020062514663, 2.14928711954148034554310854156, 3.52192649435449321981274035567, 4.52625526011822258667280086553, 6.05120035480725771543282038318, 7.01598564993259916972914767975, 7.82691177802607610854773959211, 8.710062309449484082939250070180, 9.394295258830305089581961707551, 10.14416590881543937337458946076
