L(s) = 1 | − 2-s − 4-s − 5-s − 2·7-s + 3·8-s − 3·9-s + 10-s − 6·11-s + 2·13-s + 2·14-s − 16-s − 2·17-s + 3·18-s − 2·19-s + 20-s + 6·22-s + 2·23-s + 25-s − 2·26-s + 2·28-s − 29-s + 2·31-s − 5·32-s + 2·34-s + 2·35-s + 3·36-s + 10·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s + 1.06·8-s − 9-s + 0.316·10-s − 1.80·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.458·19-s + 0.223·20-s + 1.27·22-s + 0.417·23-s + 1/5·25-s − 0.392·26-s + 0.377·28-s − 0.185·29-s + 0.359·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s + 1/2·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−12.87468148069785505472277850364, −11.24854153702415177198710204919, −10.50387677609449297409151308420, −9.378870263890314110548701917057, −8.402368279712544431210177649490, −7.64947192507391358342580089136, −6.01956800466047829467852819351, −4.64971827708119930945851978376, −2.95203182599403093553876063934, 0,
2.95203182599403093553876063934, 4.64971827708119930945851978376, 6.01956800466047829467852819351, 7.64947192507391358342580089136, 8.402368279712544431210177649490, 9.378870263890314110548701917057, 10.50387677609449297409151308420, 11.24854153702415177198710204919, 12.87468148069785505472277850364