| L(s) = 1 | + 1.40·2-s − 3·3-s − 6.01·4-s − 4.22·6-s − 13.7·7-s − 19.7·8-s + 9·9-s + 11·11-s + 18.0·12-s − 69.5·13-s − 19.3·14-s + 20.3·16-s − 48.3·17-s + 12.6·18-s − 81.6·19-s + 41.1·21-s + 15.4·22-s − 9.38·23-s + 59.2·24-s − 98.0·26-s − 27·27-s + 82.5·28-s − 10.3·29-s − 101.·31-s + 186.·32-s − 33·33-s − 68.0·34-s + ⋯ |
| L(s) = 1 | + 0.498·2-s − 0.577·3-s − 0.751·4-s − 0.287·6-s − 0.740·7-s − 0.872·8-s + 0.333·9-s + 0.301·11-s + 0.434·12-s − 1.48·13-s − 0.368·14-s + 0.317·16-s − 0.689·17-s + 0.166·18-s − 0.986·19-s + 0.427·21-s + 0.150·22-s − 0.0850·23-s + 0.503·24-s − 0.739·26-s − 0.192·27-s + 0.556·28-s − 0.0662·29-s − 0.590·31-s + 1.03·32-s − 0.174·33-s − 0.343·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6646527031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6646527031\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 1.40T + 8T^{2} \) |
| 7 | \( 1 + 13.7T + 343T^{2} \) |
| 13 | \( 1 + 69.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.38T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 81.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 420.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 176.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 42.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 237.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 569.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 957.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 295.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 752.T + 9.12e5T^{2} \) |
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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
−9.749966747574164133069909609568, −9.219547811006996007028813636626, −8.175102365191849186084553849212, −6.97677572086026135805919633580, −6.27474393649926159991684293321, −5.27106457131418682387709972363, −4.52328020802250581344159806639, −3.61722771610499500266271747120, −2.31350899245002821959872438052, −0.41318043226237735408414995671,
0.41318043226237735408414995671, 2.31350899245002821959872438052, 3.61722771610499500266271747120, 4.52328020802250581344159806639, 5.27106457131418682387709972363, 6.27474393649926159991684293321, 6.97677572086026135805919633580, 8.175102365191849186084553849212, 9.219547811006996007028813636626, 9.749966747574164133069909609568
