| L(s) = 1 | − 16·2-s − 155.·3-s + 256·4-s + 982.·5-s + 2.48e3·6-s + 5.99e3·7-s − 4.09e3·8-s + 4.38e3·9-s − 1.57e4·10-s − 1.46e4·11-s − 3.97e4·12-s − 5.59e4·13-s − 9.58e4·14-s − 1.52e5·15-s + 6.55e4·16-s − 2.05e5·17-s − 7.01e4·18-s − 2.79e5·19-s + 2.51e5·20-s − 9.29e5·21-s + 2.34e5·22-s − 1.66e6·23-s + 6.35e5·24-s − 9.88e5·25-s + 8.95e5·26-s + 2.37e6·27-s + 1.53e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s + 0.702·5-s + 0.781·6-s + 0.943·7-s − 0.353·8-s + 0.222·9-s − 0.497·10-s − 0.301·11-s − 0.552·12-s − 0.543·13-s − 0.666·14-s − 0.777·15-s + 0.250·16-s − 0.596·17-s − 0.157·18-s − 0.492·19-s + 0.351·20-s − 1.04·21-s + 0.213·22-s − 1.23·23-s + 0.390·24-s − 0.505·25-s + 0.384·26-s + 0.859·27-s + 0.471·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 16T \) |
| 11 | \( 1 + 1.46e4T \) |
| good | 3 | \( 1 + 155.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 982.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 5.99e3T + 4.03e7T^{2} \) |
| 13 | \( 1 + 5.59e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.05e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.79e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.66e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.92e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.22e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.26e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.65e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.13e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.95e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.64e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.48e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.78e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.03e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.15e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.92e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.37e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.99e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.36e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.29e8T + 7.60e17T^{2} \) |
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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
−15.56569156908112728706651063278, −13.99631597193852905423893990863, −12.17409670454820283536462162163, −11.09388809417739378899833019315, −9.985890223518318507375689218998, −8.237131069032866427488817951059, −6.44316847451934672493812962566, −5.08052989551121481348464565365, −1.90124099209667037110766883982, 0,
1.90124099209667037110766883982, 5.08052989551121481348464565365, 6.44316847451934672493812962566, 8.237131069032866427488817951059, 9.985890223518318507375689218998, 11.09388809417739378899833019315, 12.17409670454820283536462162163, 13.99631597193852905423893990863, 15.56569156908112728706651063278
