| L(s) = 1 | − 262.·3-s + 625·5-s + 1.79e3·7-s + 4.92e4·9-s − 7.42e4·11-s − 9.72e4·13-s − 1.64e5·15-s − 2.52e5·17-s − 7.26e5·19-s − 4.70e5·21-s + 1.75e6·23-s + 3.90e5·25-s − 7.77e6·27-s − 1.18e6·29-s − 6.58e6·31-s + 1.95e7·33-s + 1.11e6·35-s + 1.30e7·37-s + 2.55e7·39-s + 7.70e6·41-s + 4.14e7·43-s + 3.08e7·45-s − 2.58e7·47-s − 3.71e7·49-s + 6.61e7·51-s − 7.64e7·53-s − 4.64e7·55-s + ⋯ |
| L(s) = 1 | − 1.87·3-s + 0.447·5-s + 0.281·7-s + 2.50·9-s − 1.53·11-s − 0.944·13-s − 0.837·15-s − 0.732·17-s − 1.27·19-s − 0.527·21-s + 1.30·23-s + 0.200·25-s − 2.81·27-s − 0.312·29-s − 1.28·31-s + 2.86·33-s + 0.126·35-s + 1.14·37-s + 1.76·39-s + 0.425·41-s + 1.84·43-s + 1.11·45-s − 0.771·47-s − 0.920·49-s + 1.37·51-s − 1.33·53-s − 0.684·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6193756422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6193756422\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| good | 3 | \( 1 + 262.T + 1.96e4T^{2} \) |
| 7 | \( 1 - 1.79e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.42e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.72e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.52e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.26e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.75e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.58e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.30e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.70e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.14e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.58e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.64e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.38e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.74e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.27e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.55e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.83e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.98e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.86e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.88e8T + 7.60e17T^{2} \) |
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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
−12.70396080912768433221901678759, −11.09380559315768570319335514854, −10.77428236382325139607355177297, −9.536844592944351039172219365706, −7.61607118437349701582677039323, −6.47511628512369500305446761008, −5.34824589028862019333360751561, −4.63287395753991564806578506489, −2.16449764126106293444241531158, −0.48799360738727987325515356578,
0.48799360738727987325515356578, 2.16449764126106293444241531158, 4.63287395753991564806578506489, 5.34824589028862019333360751561, 6.47511628512369500305446761008, 7.61607118437349701582677039323, 9.536844592944351039172219365706, 10.77428236382325139607355177297, 11.09380559315768570319335514854, 12.70396080912768433221901678759
