| L(s) = 1 | + 32·2-s + 223.·3-s + 1.02e3·4-s − 6.13e3·5-s + 7.15e3·6-s − 6.71e4·7-s + 3.27e4·8-s − 1.27e5·9-s − 1.96e5·10-s − 1.61e5·11-s + 2.29e5·12-s + 2.36e5·13-s − 2.14e6·14-s − 1.37e6·15-s + 1.04e6·16-s + 1.46e6·17-s − 4.06e6·18-s − 8.85e6·19-s − 6.27e6·20-s − 1.50e7·21-s − 5.15e6·22-s + 2.18e7·23-s + 7.32e6·24-s − 1.12e7·25-s + 7.56e6·26-s − 6.80e7·27-s − 6.87e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.531·3-s + 0.5·4-s − 0.877·5-s + 0.375·6-s − 1.51·7-s + 0.353·8-s − 0.717·9-s − 0.620·10-s − 0.301·11-s + 0.265·12-s + 0.176·13-s − 1.06·14-s − 0.466·15-s + 0.250·16-s + 0.249·17-s − 0.507·18-s − 0.820·19-s − 0.438·20-s − 0.802·21-s − 0.213·22-s + 0.707·23-s + 0.187·24-s − 0.229·25-s + 0.124·26-s − 0.912·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 32T \) |
| 11 | \( 1 + 1.61e5T \) |
| good | 3 | \( 1 - 223.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 6.13e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.71e4T + 1.97e9T^{2} \) |
| 13 | \( 1 - 2.36e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.46e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.85e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.18e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 3.83e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.25e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 5.36e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.70e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.72e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.62e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.63e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 1.73e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.44e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.96e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 6.50e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 9.04e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.46e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.83e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.77e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.12e11T + 7.15e21T^{2} \) |
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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
−14.77683333704028662987915760567, −13.37607664322959305095806805924, −12.38525435850532855675697977830, −10.92589417546072243124034994866, −9.148927155801302197190547315367, −7.54089304015631796414923950284, −5.97125955092506176537552499921, −3.86837571187162189783425874570, −2.79383779579719421555766715315, 0,
2.79383779579719421555766715315, 3.86837571187162189783425874570, 5.97125955092506176537552499921, 7.54089304015631796414923950284, 9.148927155801302197190547315367, 10.92589417546072243124034994866, 12.38525435850532855675697977830, 13.37607664322959305095806805924, 14.77683333704028662987915760567
