| L(s) = 1 | − 32·2-s + 1.02e3·4-s − 9.44e3·5-s − 1.68e4·7-s − 3.27e4·8-s + 3.02e5·10-s − 6.04e5·11-s + 5.20e5·13-s + 5.37e5·14-s + 1.04e6·16-s + 8.24e6·17-s + 3.03e6·19-s − 9.66e6·20-s + 1.93e7·22-s − 1.41e7·23-s + 4.03e7·25-s − 1.66e7·26-s − 1.72e7·28-s + 1.03e8·29-s + 1.91e8·31-s − 3.35e7·32-s − 2.63e8·34-s + 1.58e8·35-s + 4.17e7·37-s − 9.72e7·38-s + 3.09e8·40-s − 1.12e9·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.35·5-s − 0.377·7-s − 0.353·8-s + 0.955·10-s − 1.13·11-s + 0.389·13-s + 0.267·14-s + 0.250·16-s + 1.40·17-s + 0.281·19-s − 0.675·20-s + 0.800·22-s − 0.458·23-s + 0.825·25-s − 0.275·26-s − 0.188·28-s + 0.934·29-s + 1.19·31-s − 0.176·32-s − 0.996·34-s + 0.510·35-s + 0.0990·37-s − 0.199·38-s + 0.477·40-s − 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 + 9.44e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 6.04e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 5.20e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 8.24e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 3.03e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.41e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.03e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.91e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.17e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.12e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.89e7T + 9.29e17T^{2} \) |
| 47 | \( 1 - 7.07e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.14e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.40e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 3.50e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 6.99e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.57e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 8.14e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 5.12e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.51e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 9.67e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.43e11T + 7.15e21T^{2} \) |
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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
−10.61496115959658580330879310468, −9.801633690063504511886619424111, −8.266016788434430759497651113863, −7.891431452427311112584734695570, −6.70489634288881745052576118120, −5.25947320691238216809862241134, −3.75306547762070004609530952621, −2.76365330972454895988839988358, −0.993686899626935506178292514018, 0,
0.993686899626935506178292514018, 2.76365330972454895988839988358, 3.75306547762070004609530952621, 5.25947320691238216809862241134, 6.70489634288881745052576118120, 7.891431452427311112584734695570, 8.266016788434430759497651113863, 9.801633690063504511886619424111, 10.61496115959658580330879310468
