L(s) = 1 | + 34.5·2-s − 81·3-s + 678.·4-s − 2.12e3·5-s − 2.79e3·6-s − 1.10e4·7-s + 5.73e3·8-s + 6.56e3·9-s − 7.33e4·10-s − 1.28e4·11-s − 5.49e4·12-s − 9.83e4·13-s − 3.82e5·14-s + 1.72e5·15-s − 1.49e5·16-s − 5.52e4·17-s + 2.26e5·18-s + 1.34e5·19-s − 1.44e6·20-s + 8.97e5·21-s − 4.43e5·22-s + 1.76e6·23-s − 4.64e5·24-s + 2.56e6·25-s − 3.39e6·26-s − 5.31e5·27-s − 7.51e6·28-s + ⋯ |
L(s) = 1 | + 1.52·2-s − 0.577·3-s + 1.32·4-s − 1.52·5-s − 0.880·6-s − 1.74·7-s + 0.495·8-s + 0.333·9-s − 2.31·10-s − 0.264·11-s − 0.764·12-s − 0.955·13-s − 2.66·14-s + 0.878·15-s − 0.569·16-s − 0.160·17-s + 0.508·18-s + 0.237·19-s − 2.01·20-s + 1.00·21-s − 0.403·22-s + 1.31·23-s − 0.285·24-s + 1.31·25-s − 1.45·26-s − 0.192·27-s − 2.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.9388169028\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9388169028\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
good | 2 | \( 1 - 34.5T + 512T^{2} \) |
| 5 | \( 1 + 2.12e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.10e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.28e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.83e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.52e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.76e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.27e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.55e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.12e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 9.02e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.12e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.08e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.63e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 5.05e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.08e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.83e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.02e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.85e6T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.30e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.42e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.45e8T + 7.60e17T^{2} \) |
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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
−11.50356430083815296393257159101, −10.37898279758439147679465438019, −9.061139593604756211835927577864, −7.31516047979433028328918022303, −6.76155873166727362092848725248, −5.54276062124378279630341976296, −4.51158270254807181681672150337, −3.55825495376209152323998613044, −2.81476950633213188528617792915, −0.36443002031314907851036081476,
0.36443002031314907851036081476, 2.81476950633213188528617792915, 3.55825495376209152323998613044, 4.51158270254807181681672150337, 5.54276062124378279630341976296, 6.76155873166727362092848725248, 7.31516047979433028328918022303, 9.061139593604756211835927577864, 10.37898279758439147679465438019, 11.50356430083815296393257159101