| L(s) = 1 | + 164.·5-s − 1.68e4·7-s + 5.65e4·11-s + 1.00e6·13-s − 3.59e6·17-s − 6.13e6·19-s + 4.65e7·23-s − 4.88e7·25-s − 1.58e8·29-s + 1.96e8·31-s − 2.77e6·35-s + 1.25e8·37-s − 8.43e7·41-s + 2.72e8·43-s + 2.86e9·47-s + 2.82e8·49-s − 3.58e9·53-s + 9.32e6·55-s − 5.45e9·59-s − 2.53e9·61-s + 1.65e8·65-s − 1.58e10·67-s + 2.55e10·71-s + 3.01e10·73-s − 9.51e8·77-s − 7.76e9·79-s − 6.93e10·83-s + ⋯ |
| L(s) = 1 | + 0.0235·5-s − 0.377·7-s + 0.105·11-s + 0.749·13-s − 0.614·17-s − 0.568·19-s + 1.50·23-s − 0.999·25-s − 1.43·29-s + 1.23·31-s − 0.00891·35-s + 0.297·37-s − 0.113·41-s + 0.282·43-s + 1.82·47-s + 0.142·49-s − 1.17·53-s + 0.00249·55-s − 0.993·59-s − 0.383·61-s + 0.0176·65-s − 1.43·67-s + 1.68·71-s + 1.70·73-s − 0.0400·77-s − 0.283·79-s − 1.93·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 - 164.T + 4.88e7T^{2} \) |
| 11 | \( 1 - 5.65e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.00e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.59e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 6.13e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.65e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.58e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.96e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.25e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 8.43e7T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.72e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.86e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.58e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 5.45e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 2.53e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.58e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.55e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.01e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 7.76e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.93e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.39e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.27e10T + 7.15e21T^{2} \) |
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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
−9.508359605135730896795737232204, −8.795815475692363665526163993730, −7.68332079598069403620323753667, −6.62028850791825973304646860667, −5.78875548200696046178182517444, −4.52081072551261687832771589034, −3.52477124360133404995846030587, −2.37622132284294517638795082628, −1.17035140245095316552406945632, 0,
1.17035140245095316552406945632, 2.37622132284294517638795082628, 3.52477124360133404995846030587, 4.52081072551261687832771589034, 5.78875548200696046178182517444, 6.62028850791825973304646860667, 7.68332079598069403620323753667, 8.795815475692363665526163993730, 9.508359605135730896795737232204
