| L(s) = 1 | + 24.7·2-s + 99.0·4-s + 920.·5-s + 5.35e3·7-s − 1.02e4·8-s + 2.27e4·10-s − 7.92e4·11-s + 2.85e4·13-s + 1.32e5·14-s − 3.03e5·16-s − 4.52e5·17-s + 2.12e5·19-s + 9.11e4·20-s − 1.95e6·22-s + 7.59e5·23-s − 1.10e6·25-s + 7.05e5·26-s + 5.30e5·28-s + 9.00e5·29-s + 2.27e6·31-s − 2.26e6·32-s − 1.11e7·34-s + 4.93e6·35-s − 4.70e6·37-s + 5.25e6·38-s − 9.39e6·40-s − 3.39e7·41-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.193·4-s + 0.658·5-s + 0.843·7-s − 0.881·8-s + 0.719·10-s − 1.63·11-s + 0.277·13-s + 0.921·14-s − 1.15·16-s − 1.31·17-s + 0.374·19-s + 0.127·20-s − 1.78·22-s + 0.565·23-s − 0.566·25-s + 0.302·26-s + 0.163·28-s + 0.236·29-s + 0.441·31-s − 0.381·32-s − 1.43·34-s + 0.555·35-s − 0.412·37-s + 0.408·38-s − 0.580·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 24.7T + 512T^{2} \) |
| 5 | \( 1 - 920.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.35e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.92e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 4.52e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.12e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.59e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.00e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.70e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.39e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.33e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.14e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.01e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.23e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.15e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.06e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.44e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.03e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.20e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.17e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.91e8T + 7.60e17T^{2} \) |
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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
−11.39079364067237090914526783406, −10.40117496293547387382045766105, −9.067422505885045694982034140839, −7.962264470198401350605350622970, −6.43146944984247054531534436405, −5.27334035077340168717428421419, −4.66760598741898621022299269417, −3.08144236756302089480049040464, −1.92430050765043199594857755779, 0,
1.92430050765043199594857755779, 3.08144236756302089480049040464, 4.66760598741898621022299269417, 5.27334035077340168717428421419, 6.43146944984247054531534436405, 7.962264470198401350605350622970, 9.067422505885045694982034140839, 10.40117496293547387382045766105, 11.39079364067237090914526783406
