| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 3·5-s − 2·6-s + 5·7-s + 9-s + 6·10-s − 11-s − 2·12-s + 10·14-s − 3·15-s − 4·16-s − 17-s + 2·18-s + 19-s + 6·20-s − 5·21-s − 2·22-s − 4·23-s + 4·25-s − 27-s + 10·28-s − 2·29-s − 6·30-s + 6·31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 1.34·5-s − 0.816·6-s + 1.88·7-s + 1/3·9-s + 1.89·10-s − 0.301·11-s − 0.577·12-s + 2.67·14-s − 0.774·15-s − 16-s − 0.242·17-s + 0.471·18-s + 0.229·19-s + 1.34·20-s − 1.09·21-s − 0.426·22-s − 0.834·23-s + 4/5·25-s − 0.192·27-s + 1.88·28-s − 0.371·29-s − 1.09·30-s + 1.07·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.163278896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.163278896\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.47907135182185058330172781462, −6.67667593870166550135695113551, −5.95144563794148553465287151472, −5.50249705484339129095277632117, −5.02848669373136257535239205159, −4.46569280630286556622369413616, −3.73362270097943231685814461481, −2.36240806997936697850725908993, −2.12263958916626507979414707605, −1.03092043898145556677704805507,
1.03092043898145556677704805507, 2.12263958916626507979414707605, 2.36240806997936697850725908993, 3.73362270097943231685814461481, 4.46569280630286556622369413616, 5.02848669373136257535239205159, 5.50249705484339129095277632117, 5.95144563794148553465287151472, 6.67667593870166550135695113551, 7.47907135182185058330172781462
