| L(s) = 1 | + 34.8·2-s + 703.·4-s − 2.00e3·5-s + 4.86e3·7-s + 6.68e3·8-s − 6.99e4·10-s + 7.39e4·11-s + 2.64e4·13-s + 1.69e5·14-s − 1.27e5·16-s + 3.00e5·17-s + 1.80e5·19-s − 1.41e6·20-s + 2.57e6·22-s + 2.13e6·23-s + 2.07e6·25-s + 9.23e5·26-s + 3.42e6·28-s + 1.89e6·29-s + 5.07e6·31-s − 7.85e6·32-s + 1.04e7·34-s − 9.76e6·35-s − 2.17e6·37-s + 6.29e6·38-s − 1.34e7·40-s + 7.63e6·41-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.37·4-s − 1.43·5-s + 0.766·7-s + 0.576·8-s − 2.21·10-s + 1.52·11-s + 0.257·13-s + 1.18·14-s − 0.485·16-s + 0.872·17-s + 0.317·19-s − 1.97·20-s + 2.34·22-s + 1.58·23-s + 1.06·25-s + 0.396·26-s + 1.05·28-s + 0.497·29-s + 0.986·31-s − 1.32·32-s + 1.34·34-s − 1.10·35-s − 0.190·37-s + 0.489·38-s − 0.828·40-s + 0.421·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.680933651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.680933651\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 34.8T + 512T^{2} \) |
| 5 | \( 1 + 2.00e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.86e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.39e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.64e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.00e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.80e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.13e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.89e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.07e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.17e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.63e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.37e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.60e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.22e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.87e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.71e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 8.48e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.62e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.90e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.26e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.61e9T + 7.60e17T^{2} \) |
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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
−12.27818478731238687362975908561, −11.81539883803416648308359987858, −11.00304527554797982108211713742, −8.945114310164150626121909076779, −7.63251997826544929638180490211, −6.49098664691017065755002412613, −4.97343962532240215808624884004, −4.07440119783667610187292103456, −3.15986725694404111905442763365, −1.10118942545867924182290363623,
1.10118942545867924182290363623, 3.15986725694404111905442763365, 4.07440119783667610187292103456, 4.97343962532240215808624884004, 6.49098664691017065755002412613, 7.63251997826544929638180490211, 8.945114310164150626121909076779, 11.00304527554797982108211713742, 11.81539883803416648308359987858, 12.27818478731238687362975908561
