| L(s) = 1 | − 9·3-s + 27.8·5-s − 58.7·7-s + 81·9-s + 130.·11-s − 150.·13-s − 250.·15-s + 1.15e3·17-s − 2.42e3·19-s + 528.·21-s − 529·23-s − 2.35e3·25-s − 729·27-s + 1.22e3·29-s + 5.71e3·31-s − 1.17e3·33-s − 1.63e3·35-s + 1.62e4·37-s + 1.35e3·39-s − 2.31e3·41-s − 1.28e4·43-s + 2.25e3·45-s + 524.·47-s − 1.33e4·49-s − 1.03e4·51-s + 3.32e4·53-s + 3.62e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.497·5-s − 0.452·7-s + 0.333·9-s + 0.324·11-s − 0.246·13-s − 0.287·15-s + 0.968·17-s − 1.53·19-s + 0.261·21-s − 0.208·23-s − 0.752·25-s − 0.192·27-s + 0.269·29-s + 1.06·31-s − 0.187·33-s − 0.225·35-s + 1.95·37-s + 0.142·39-s − 0.215·41-s − 1.05·43-s + 0.165·45-s + 0.0346·47-s − 0.794·49-s − 0.559·51-s + 1.62·53-s + 0.161·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.583254306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.583254306\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 - 27.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 58.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 130.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 150.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.15e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.42e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 1.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.62e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.31e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 524.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.79e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.10e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.29e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.62e4T + 8.58e9T^{2} \) |
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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
−9.276306211291138261143820373050, −8.297573036320281179444032544211, −7.39606297653066122406039526493, −6.27988217758638470367275840606, −6.03771964411779910433483011822, −4.84842439906302518285629174651, −3.99724969536709130602228522128, −2.79023216684920880009245540685, −1.69977240849134119625022787492, −0.55629207801729246623391765380,
0.55629207801729246623391765380, 1.69977240849134119625022787492, 2.79023216684920880009245540685, 3.99724969536709130602228522128, 4.84842439906302518285629174651, 6.03771964411779910433483011822, 6.27988217758638470367275840606, 7.39606297653066122406039526493, 8.297573036320281179444032544211, 9.276306211291138261143820373050
