| L(s) = 1 | − 26.9·2-s + 724.·3-s − 1.32e3·4-s − 9.76e3·5-s − 1.95e4·6-s − 7.30e4·7-s + 9.08e4·8-s + 3.48e5·9-s + 2.63e5·10-s + 5.27e5·11-s − 9.57e5·12-s + 9.61e5·13-s + 1.96e6·14-s − 7.07e6·15-s + 2.55e5·16-s + 1.07e7·17-s − 9.39e6·18-s − 7.28e6·19-s + 1.28e7·20-s − 5.29e7·21-s − 1.42e7·22-s + 3.19e7·23-s + 6.58e7·24-s + 4.64e7·25-s − 2.59e7·26-s + 1.24e8·27-s + 9.64e7·28-s + ⋯ |
| L(s) = 1 | − 0.595·2-s + 1.72·3-s − 0.644·4-s − 1.39·5-s − 1.02·6-s − 1.64·7-s + 0.980·8-s + 1.96·9-s + 0.832·10-s + 0.986·11-s − 1.11·12-s + 0.717·13-s + 0.978·14-s − 2.40·15-s + 0.0609·16-s + 1.82·17-s − 1.17·18-s − 0.675·19-s + 0.901·20-s − 2.82·21-s − 0.588·22-s + 1.03·23-s + 1.68·24-s + 0.951·25-s − 0.427·26-s + 1.66·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.541542139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541542139\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 + 26.9T + 2.04e3T^{2} \) |
| 3 | \( 1 - 724.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 9.76e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 7.30e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.27e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 9.61e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.07e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 7.28e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.19e7T + 9.52e14T^{2} \) |
| 31 | \( 1 - 1.82e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.13e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.96e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 5.26e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 9.33e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.28e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 2.52e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.85e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.28e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.02e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 5.32e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.32e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 9.76e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 8.30e9T + 2.77e21T^{2} \) |
| 97 | \( 1 - 7.76e9T + 7.15e21T^{2} \) |
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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
−14.58678388154991378595237456032, −13.44867026869536664797585209967, −12.36419482147913643687114463964, −10.11901505981129084467353104088, −9.086897477454382996570888439551, −8.278560684366518569249447254856, −7.09622331257722787764059565955, −3.88749032109176227649526946740, −3.34187228697509942475865957520, −0.878272588675919680788110847456,
0.878272588675919680788110847456, 3.34187228697509942475865957520, 3.88749032109176227649526946740, 7.09622331257722787764059565955, 8.278560684366518569249447254856, 9.086897477454382996570888439551, 10.11901505981129084467353104088, 12.36419482147913643687114463964, 13.44867026869536664797585209967, 14.58678388154991378595237456032
