L(s) = 1 | − 625.·2-s + 5.90e4·3-s − 1.70e6·4-s − 3.69e7·6-s − 3.78e8·7-s + 2.37e9·8-s + 3.48e9·9-s − 3.73e10·11-s − 1.00e11·12-s + 9.27e11·13-s + 2.36e11·14-s + 2.09e12·16-s + 1.40e13·17-s − 2.18e12·18-s + 3.31e13·19-s − 2.23e13·21-s + 2.33e13·22-s − 3.30e14·23-s + 1.40e14·24-s − 5.79e14·26-s + 2.05e14·27-s + 6.46e14·28-s + 2.35e15·29-s + 3.97e15·31-s − 6.29e15·32-s − 2.20e15·33-s − 8.77e15·34-s + ⋯ |
L(s) = 1 | − 0.431·2-s + 0.577·3-s − 0.813·4-s − 0.249·6-s − 0.506·7-s + 0.783·8-s + 0.333·9-s − 0.433·11-s − 0.469·12-s + 1.86·13-s + 0.218·14-s + 0.475·16-s + 1.68·17-s − 0.143·18-s + 1.23·19-s − 0.292·21-s + 0.187·22-s − 1.66·23-s + 0.452·24-s − 0.805·26-s + 0.192·27-s + 0.412·28-s + 1.04·29-s + 0.870·31-s − 0.988·32-s − 0.250·33-s − 0.729·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(2.132002004\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.132002004\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 625.T + 2.09e6T^{2} \) |
| 7 | \( 1 + 3.78e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.73e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 9.27e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.40e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.31e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.30e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.35e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 3.97e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.11e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 5.23e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.10e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 5.77e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.16e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 4.17e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 9.77e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.53e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 6.75e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.58e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.48e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 6.30e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.22e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 5.41e20T + 5.27e41T^{2} \) |
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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
−10.06554428549136163462493952707, −9.754050224771117892076275995468, −8.282337802011442035470211316656, −8.018814247650825381964538358578, −6.35386830467607815544680369770, −5.19237455553997743869634004930, −3.81276074114955752671402312844, −3.15877610266214529398002195473, −1.46592929815049956714237597212, −0.70407349876262103304756547841,
0.70407349876262103304756547841, 1.46592929815049956714237597212, 3.15877610266214529398002195473, 3.81276074114955752671402312844, 5.19237455553997743869634004930, 6.35386830467607815544680369770, 8.018814247650825381964538358578, 8.282337802011442035470211316656, 9.754050224771117892076275995468, 10.06554428549136163462493952707