| L(s) = 1 | − 128·2-s + 1.35e3·3-s + 1.63e4·4-s − 8.10e4·5-s − 1.72e5·6-s + 8.23e5·7-s − 2.09e6·8-s − 1.25e7·9-s + 1.03e7·10-s + 7.01e7·11-s + 2.21e7·12-s + 1.51e8·13-s − 1.05e8·14-s − 1.09e8·15-s + 2.68e8·16-s − 2.49e8·17-s + 1.60e9·18-s − 6.47e9·19-s − 1.32e9·20-s + 1.11e9·21-s − 8.97e9·22-s − 2.11e10·23-s − 2.83e9·24-s − 2.39e10·25-s − 1.93e10·26-s − 3.62e10·27-s + 1.34e10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.356·3-s + 1/2·4-s − 0.464·5-s − 0.252·6-s + 0.377·7-s − 0.353·8-s − 0.872·9-s + 0.328·10-s + 1.08·11-s + 0.178·12-s + 0.669·13-s − 0.267·14-s − 0.165·15-s + 1/4·16-s − 0.147·17-s + 0.617·18-s − 1.66·19-s − 0.232·20-s + 0.134·21-s − 0.767·22-s − 1.29·23-s − 0.126·24-s − 0.784·25-s − 0.473·26-s − 0.667·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{7} T \) |
| 7 | \( 1 - p^{7} T \) |
| good | 3 | \( 1 - 50 p^{3} T + p^{15} T^{2} \) |
| 5 | \( 1 + 16212 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 70121184 T + p^{15} T^{2} \) |
| 13 | \( 1 - 11651504 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 249756546 T + p^{15} T^{2} \) |
| 19 | \( 1 + 6476856550 T + p^{15} T^{2} \) |
| 23 | \( 1 + 21129196200 T + p^{15} T^{2} \) |
| 29 | \( 1 - 7794825354 T + p^{15} T^{2} \) |
| 31 | \( 1 + 95032053412 T + p^{15} T^{2} \) |
| 37 | \( 1 + 870082295470 T + p^{15} T^{2} \) |
| 41 | \( 1 - 1007666657262 T + p^{15} T^{2} \) |
| 43 | \( 1 - 155007585272 T + p^{15} T^{2} \) |
| 47 | \( 1 + 2551970135004 T + p^{15} T^{2} \) |
| 53 | \( 1 - 4047645687774 T + p^{15} T^{2} \) |
| 59 | \( 1 + 12599248786302 T + p^{15} T^{2} \) |
| 61 | \( 1 + 39925031318044 T + p^{15} T^{2} \) |
| 67 | \( 1 + 722742988972 p T + p^{15} T^{2} \) |
| 71 | \( 1 - 37693101366144 T + p^{15} T^{2} \) |
| 73 | \( 1 - 141416194574306 T + p^{15} T^{2} \) |
| 79 | \( 1 - 247020521013128 T + p^{15} T^{2} \) |
| 83 | \( 1 - 2788789610034 T + p^{15} T^{2} \) |
| 89 | \( 1 + 5839634731110 T + p^{15} T^{2} \) |
| 97 | \( 1 - 278027158065374 T + p^{15} T^{2} \) |
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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
−15.24445050721253091849666997240, −14.05454769907852157179429392773, −12.02972085193207668166112594239, −10.86964536124684821067430099283, −9.029425696969759913700715553322, −8.050093527222915500640925081182, −6.24384659130351074005963841213, −3.82522755711110525882851405345, −1.88950506932279134772583527417, 0,
1.88950506932279134772583527417, 3.82522755711110525882851405345, 6.24384659130351074005963841213, 8.050093527222915500640925081182, 9.029425696969759913700715553322, 10.86964536124684821067430099283, 12.02972085193207668166112594239, 14.05454769907852157179429392773, 15.24445050721253091849666997240
