| L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s − 3·8-s + 9-s − 2·10-s + 4·11-s − 12-s − 13-s − 2·15-s − 16-s + 17-s + 18-s + 2·20-s + 4·22-s − 3·24-s − 25-s − 26-s + 27-s + 2·29-s − 2·30-s + 8·31-s + 5·32-s + 4·33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.516·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.447·20-s + 0.852·22-s − 0.612·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s + 0.696·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 239343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 239343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.636983978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.636983978\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.89573425812871, −12.49460287661151, −11.92526005415488, −11.67363247892367, −11.42578360992120, −10.44161047209226, −10.04499887373545, −9.625181317990838, −9.068422608275889, −8.584591846177923, −8.353899996458535, −7.716005604801913, −7.243449123735402, −6.654082268979444, −6.205410060581887, −5.672748879226257, −4.885734688965842, −4.618953393731036, −3.962358152355287, −3.769099499641569, −3.186384081355906, −2.642223847309480, −1.925150220065968, −1.016264229799590, −0.5325552882795344,
0.5325552882795344, 1.016264229799590, 1.925150220065968, 2.642223847309480, 3.186384081355906, 3.769099499641569, 3.962358152355287, 4.618953393731036, 4.885734688965842, 5.672748879226257, 6.205410060581887, 6.654082268979444, 7.243449123735402, 7.716005604801913, 8.353899996458535, 8.584591846177923, 9.068422608275889, 9.625181317990838, 10.04499887373545, 10.44161047209226, 11.42578360992120, 11.67363247892367, 11.92526005415488, 12.49460287661151, 12.89573425812871
