| L(s) = 1 | + 97·2-s + 5.31e3·4-s + 2.92e4·5-s − 5.03e4·7-s + 1.18e5·8-s + 5.31e5·9-s + 2.83e6·10-s − 4.88e6·14-s − 1.03e7·16-s + 5.15e7·18-s + 1.86e7·19-s + 1.55e8·20-s + 6.12e8·25-s − 2.67e8·28-s + 8.87e8·31-s − 1.48e9·32-s − 1.47e9·35-s + 2.82e9·36-s + 1.80e9·38-s + 3.45e9·40-s − 5.83e9·41-s + 1.55e10·45-s + 7.96e9·47-s − 1.13e10·49-s + 5.93e10·50-s − 5.94e9·56-s − 6.56e10·59-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.29·4-s + 1.87·5-s − 0.428·7-s + 0.450·8-s + 9-s + 2.83·10-s − 0.649·14-s − 0.614·16-s + 1.51·18-s + 0.396·19-s + 2.42·20-s + 2.50·25-s − 0.555·28-s + 31-s − 1.38·32-s − 0.802·35-s + 1.29·36-s + 0.600·38-s + 0.843·40-s − 1.22·41-s + 1.87·45-s + 0.738·47-s − 0.816·49-s + 3.80·50-s − 0.192·56-s − 1.55·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(6.199565984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.199565984\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 - p^{6} T \) |
| good | 2 | \( 1 - 97 T + p^{12} T^{2} \) |
| 3 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 5 | \( 1 - 29266 T + p^{12} T^{2} \) |
| 7 | \( 1 + 50398 T + p^{12} T^{2} \) |
| 11 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 13 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 17 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 19 | \( 1 - 18650162 T + p^{12} T^{2} \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 37 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 41 | \( 1 + 5832937118 T + p^{12} T^{2} \) |
| 43 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 47 | \( 1 - 7960245442 T + p^{12} T^{2} \) |
| 53 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 59 | \( 1 + 65635334318 T + p^{12} T^{2} \) |
| 61 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 67 | \( 1 + 163049095438 T + p^{12} T^{2} \) |
| 71 | \( 1 + 175443432158 T + p^{12} T^{2} \) |
| 73 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 79 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 83 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 89 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 97 | \( 1 - 1601076090242 T + p^{12} T^{2} \) |
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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
−13.75328513806506796312709347707, −13.29480697601699309174747576576, −12.25213639546959925837410473330, −10.40047624323426198151318492465, −9.354213755740272463729800126823, −6.78450523155803507121879541465, −5.85126863394289819991167382666, −4.67670584451918799826720606738, −2.95255408877476123530271839540, −1.62150884721752149568212367164,
1.62150884721752149568212367164, 2.95255408877476123530271839540, 4.67670584451918799826720606738, 5.85126863394289819991167382666, 6.78450523155803507121879541465, 9.354213755740272463729800126823, 10.40047624323426198151318492465, 12.25213639546959925837410473330, 13.29480697601699309174747576576, 13.75328513806506796312709347707
