| L(s) = 1 | − 3-s − 2·4-s − 2·5-s + 7-s − 2·9-s + 2·12-s − 4·13-s + 2·15-s + 4·16-s + 17-s − 7·19-s + 4·20-s − 21-s − 25-s + 5·27-s − 2·28-s + 29-s − 3·31-s − 2·35-s + 4·36-s + 8·37-s + 4·39-s + 6·41-s − 12·43-s + 4·45-s + 8·47-s − 4·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.894·5-s + 0.377·7-s − 2/3·9-s + 0.577·12-s − 1.10·13-s + 0.516·15-s + 16-s + 0.242·17-s − 1.60·19-s + 0.894·20-s − 0.218·21-s − 1/5·25-s + 0.962·27-s − 0.377·28-s + 0.185·29-s − 0.538·31-s − 0.338·35-s + 2/3·36-s + 1.31·37-s + 0.640·39-s + 0.937·41-s − 1.82·43-s + 0.596·45-s + 1.16·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24563 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.96334432651637, −15.24859445486459, −14.64136605787682, −14.59744492037152, −13.84871156223540, −13.11042910015609, −12.62186919798999, −12.18903200799194, −11.57986293207936, −11.19650540326679, −10.44947002135936, −10.04299695347874, −9.223103525953882, −8.753996307577040, −8.118668804970868, −7.777119641769168, −7.052096940810046, −6.224701513934032, −5.649059389510658, −5.035437307792762, −4.369728820029688, −4.112336344887365, −3.125003439442079, −2.406190078741217, −1.248007568182868, 0, 0,
1.248007568182868, 2.406190078741217, 3.125003439442079, 4.112336344887365, 4.369728820029688, 5.035437307792762, 5.649059389510658, 6.224701513934032, 7.052096940810046, 7.777119641769168, 8.118668804970868, 8.753996307577040, 9.223103525953882, 10.04299695347874, 10.44947002135936, 11.19650540326679, 11.57986293207936, 12.18903200799194, 12.62186919798999, 13.11042910015609, 13.84871156223540, 14.59744492037152, 14.64136605787682, 15.24859445486459, 15.96334432651637
