| L(s) = 1 | + 22.3·2-s + 27·3-s + 372.·4-s − 118.·5-s + 604.·6-s + 5.46e3·8-s + 729·9-s − 2.66e3·10-s − 3.52e3·11-s + 1.00e4·12-s + 5.25e3·13-s − 3.21e3·15-s + 7.46e4·16-s + 3.70e4·17-s + 1.63e4·18-s − 5.16e3·19-s − 4.43e4·20-s − 7.87e4·22-s + 3.45e4·23-s + 1.47e5·24-s − 6.39e4·25-s + 1.17e5·26-s + 1.96e4·27-s + 6.38e3·29-s − 7.18e4·30-s − 1.55e5·31-s + 9.70e5·32-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 0.577·3-s + 2.90·4-s − 0.425·5-s + 1.14·6-s + 3.77·8-s + 0.333·9-s − 0.841·10-s − 0.797·11-s + 1.67·12-s + 0.663·13-s − 0.245·15-s + 4.55·16-s + 1.82·17-s + 0.659·18-s − 0.172·19-s − 1.23·20-s − 1.57·22-s + 0.592·23-s + 2.18·24-s − 0.818·25-s + 1.31·26-s + 0.192·27-s + 0.0485·29-s − 0.485·30-s − 0.939·31-s + 5.23·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(9.302296720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.302296720\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 22.3T + 128T^{2} \) |
| 5 | \( 1 + 118.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.25e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.70e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.16e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.45e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.38e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.55e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.87e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.61e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.34e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.08e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.04e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.34e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.06e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.72e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.24e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.67e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.16e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.59e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.06e7T + 8.07e13T^{2} \) |
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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
−12.08059440061763719967618251609, −11.10802455526449982262046149862, −10.09941412871488634419243244398, −8.065575589370464190174585435158, −7.30430309943132874935520895833, −5.96816082110429466691460646983, −4.98337428488616626877340636288, −3.71435913299227066128636141832, −3.01243567119343944294351983441, −1.58218103126525913990688271400,
1.58218103126525913990688271400, 3.01243567119343944294351983441, 3.71435913299227066128636141832, 4.98337428488616626877340636288, 5.96816082110429466691460646983, 7.30430309943132874935520895833, 8.065575589370464190174585435158, 10.09941412871488634419243244398, 11.10802455526449982262046149862, 12.08059440061763719967618251609
