| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s − 4·13-s − 14-s + 16-s − 2·17-s − 18-s + 21-s + 22-s + 23-s − 24-s − 5·25-s + 4·26-s + 27-s + 28-s + 10·29-s − 6·31-s − 32-s − 33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + 1.85·29-s − 1.07·31-s − 0.176·32-s − 0.174·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| 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
−13.05688231326294, −12.43831295254048, −11.99326062656265, −11.70821939051399, −11.15107837173938, −10.56621681684309, −10.14000119737814, −9.943239949979279, −9.275864586447146, −8.922917821460167, −8.391738955986354, −8.066320580018225, −7.565236775801995, −7.141898820444298, −6.678305973735893, −6.214661017867373, −5.451744847588939, −4.993965199041587, −4.579768251964178, −3.919630928565419, −3.289251155681671, −2.739128778023086, −2.344468818253969, −1.619183749987448, −1.317666180625187, 0, 0,
1.317666180625187, 1.619183749987448, 2.344468818253969, 2.739128778023086, 3.289251155681671, 3.919630928565419, 4.579768251964178, 4.993965199041587, 5.451744847588939, 6.214661017867373, 6.678305973735893, 7.141898820444298, 7.565236775801995, 8.066320580018225, 8.391738955986354, 8.922917821460167, 9.275864586447146, 9.943239949979279, 10.14000119737814, 10.56621681684309, 11.15107837173938, 11.70821939051399, 11.99326062656265, 12.43831295254048, 13.05688231326294
