| L(s) = 1 | + 2·2-s + 2·4-s + 5-s − 2·7-s + 2·10-s − 11-s + 2·13-s − 4·14-s − 4·16-s − 2·17-s + 2·20-s − 2·22-s + 4·23-s + 25-s + 4·26-s − 4·28-s + 5·29-s + 9·31-s − 8·32-s − 4·34-s − 2·35-s − 6·37-s − 6·41-s − 10·43-s − 2·44-s + 8·46-s − 3·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s + 0.632·10-s − 0.301·11-s + 0.554·13-s − 1.06·14-s − 16-s − 0.485·17-s + 0.447·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.784·26-s − 0.755·28-s + 0.928·29-s + 1.61·31-s − 1.41·32-s − 0.685·34-s − 0.338·35-s − 0.986·37-s − 0.937·41-s − 1.52·43-s − 0.301·44-s + 1.17·46-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.88916821064774, −15.49568639494830, −15.22820948132853, −14.34394402837085, −13.87682315461369, −13.46491456378255, −13.02760572883278, −12.55405963888388, −11.88438441663771, −11.44269228474228, −10.65808195401795, −10.10281234128630, −9.533204285210794, −8.660498527260020, −8.402303343010062, −7.178960331395390, −6.654950678715853, −6.251723949669223, −5.616524913783952, −4.836086910142212, −4.523888741426811, −3.449824645287124, −3.115748464352751, −2.395648367665589, −1.360015620797932, 0,
1.360015620797932, 2.395648367665589, 3.115748464352751, 3.449824645287124, 4.523888741426811, 4.836086910142212, 5.616524913783952, 6.251723949669223, 6.654950678715853, 7.178960331395390, 8.402303343010062, 8.660498527260020, 9.533204285210794, 10.10281234128630, 10.65808195401795, 11.44269228474228, 11.88438441663771, 12.55405963888388, 13.02760572883278, 13.46491456378255, 13.87682315461369, 14.34394402837085, 15.22820948132853, 15.49568639494830, 15.88916821064774
