| L(s) = 1 | − 5-s − 2·7-s − 4·11-s + 13-s + 17-s − 8·23-s + 25-s − 6·29-s − 2·31-s + 2·35-s + 4·37-s + 2·41-s − 4·43-s − 2·47-s − 3·49-s − 10·53-s + 4·55-s − 10·59-s + 14·61-s − 65-s − 12·67-s + 8·71-s + 8·73-s + 8·77-s − 14·83-s − 85-s + 8·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.277·13-s + 0.242·17-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.359·31-s + 0.338·35-s + 0.657·37-s + 0.312·41-s − 0.609·43-s − 0.291·47-s − 3/7·49-s − 1.37·53-s + 0.539·55-s − 1.30·59-s + 1.79·61-s − 0.124·65-s − 1.46·67-s + 0.949·71-s + 0.936·73-s + 0.911·77-s − 1.53·83-s − 0.108·85-s + 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.52708585611030, −13.93702267540711, −13.44858370074809, −12.89599134519234, −12.66782381949715, −12.07232431429780, −11.46147070934899, −11.05020387121208, −10.46737869356041, −9.998024165610515, −9.526698225845596, −9.047542817695030, −8.219335379377848, −7.860382524364253, −7.597124203755278, −6.673693166203935, −6.366716426555258, −5.597949121971201, −5.292992178286031, −4.447378893651424, −3.898019263474539, −3.355718021193605, −2.754258154886445, −2.093076968312775, −1.295830753813335, 0, 0,
1.295830753813335, 2.093076968312775, 2.754258154886445, 3.355718021193605, 3.898019263474539, 4.447378893651424, 5.292992178286031, 5.597949121971201, 6.366716426555258, 6.673693166203935, 7.597124203755278, 7.860382524364253, 8.219335379377848, 9.047542817695030, 9.526698225845596, 9.998024165610515, 10.46737869356041, 11.05020387121208, 11.46147070934899, 12.07232431429780, 12.66782381949715, 12.89599134519234, 13.44858370074809, 13.93702267540711, 14.52708585611030
