| L(s) = 1 | − 29.5·3-s − 107.·5-s − 142.·7-s + 628.·9-s + 553.·11-s + 856.·13-s + 3.16e3·15-s − 974.·17-s − 726.·19-s + 4.19e3·21-s + 3.14e3·23-s + 8.36e3·25-s − 1.13e4·27-s + 1.05e3·29-s + 9.54e3·31-s − 1.63e4·33-s + 1.52e4·35-s + 1.35e4·37-s − 2.52e4·39-s − 1.38e4·41-s − 4.94e3·43-s − 6.73e4·45-s + 2.09e4·47-s + 3.43e3·49-s + 2.87e4·51-s + 1.36e4·53-s − 5.93e4·55-s + ⋯ |
| L(s) = 1 | − 1.89·3-s − 1.91·5-s − 1.09·7-s + 2.58·9-s + 1.37·11-s + 1.40·13-s + 3.63·15-s − 0.818·17-s − 0.461·19-s + 2.07·21-s + 1.23·23-s + 2.67·25-s − 2.99·27-s + 0.233·29-s + 1.78·31-s − 2.61·33-s + 2.10·35-s + 1.62·37-s − 2.66·39-s − 1.29·41-s − 0.408·43-s − 4.95·45-s + 1.38·47-s + 0.204·49-s + 1.54·51-s + 0.665·53-s − 2.64·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7416991225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7416991225\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 + 6.60e4T \) |
| good | 3 | \( 1 + 29.5T + 243T^{2} \) |
| 5 | \( 1 + 107.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 142.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 553.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 856.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 974.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 726.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.54e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.35e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.94e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 771.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.35e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.66e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.89e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 588.T + 8.58e9T^{2} \) |
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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
−9.215822210667343203817541357404, −8.363229834089297106907408450305, −7.10393753181822533420372079300, −6.60107108645089145483346392659, −6.11340834957112294248749591599, −4.64906419300099161700420668671, −4.15655503209182367769103552030, −3.34066076450369944318089934568, −1.03097922791804315092297890388, −0.55656991012825403107431546318,
0.55656991012825403107431546318, 1.03097922791804315092297890388, 3.34066076450369944318089934568, 4.15655503209182367769103552030, 4.64906419300099161700420668671, 6.11340834957112294248749591599, 6.60107108645089145483346392659, 7.10393753181822533420372079300, 8.363229834089297106907408450305, 9.215822210667343203817541357404
