| L(s) = 1 | − 5.35·2-s − 2.01e3·4-s + 1.28e4·5-s − 3.17e4·7-s + 2.17e4·8-s − 6.87e4·10-s − 6.84e5·11-s + 9.86e5·13-s + 1.69e5·14-s + 4.01e6·16-s + 8.95e6·17-s − 3.35e6·19-s − 2.59e7·20-s + 3.66e6·22-s + 1.07e7·23-s + 1.16e8·25-s − 5.27e6·26-s + 6.40e7·28-s + 1.55e8·29-s + 2.24e8·31-s − 6.60e7·32-s − 4.79e7·34-s − 4.07e8·35-s + 2.97e8·37-s + 1.79e7·38-s + 2.79e8·40-s + 3.67e8·41-s + ⋯ |
| L(s) = 1 | − 0.118·2-s − 0.986·4-s + 1.83·5-s − 0.713·7-s + 0.234·8-s − 0.217·10-s − 1.28·11-s + 0.736·13-s + 0.0844·14-s + 0.958·16-s + 1.53·17-s − 0.310·19-s − 1.81·20-s + 0.151·22-s + 0.347·23-s + 2.37·25-s − 0.0871·26-s + 0.703·28-s + 1.40·29-s + 1.40·31-s − 0.348·32-s − 0.180·34-s − 1.31·35-s + 0.705·37-s + 0.0367·38-s + 0.431·40-s + 0.494·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.843114736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.843114736\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 5.35T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.28e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 3.17e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 6.84e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 9.86e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 8.95e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 3.35e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.07e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.55e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.24e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.97e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 3.67e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.37e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.81e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.49e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 5.25e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 3.90e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.59e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.95e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 9.74e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.34e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.63e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.64e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 2.75e10T + 7.15e21T^{2} \) |
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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
−14.38405473958341063448838772395, −13.43120580030850110149395658513, −12.80232763869813051607130320574, −10.26324127072240721683429364058, −9.768754067030390856375991817041, −8.354110084275208032013223108904, −6.18069329241396162096689526656, −5.11398595726919526980996532057, −2.89481837802962152342669952903, −1.00732906819633023293780544539,
1.00732906819633023293780544539, 2.89481837802962152342669952903, 5.11398595726919526980996532057, 6.18069329241396162096689526656, 8.354110084275208032013223108904, 9.768754067030390856375991817041, 10.26324127072240721683429364058, 12.80232763869813051607130320574, 13.43120580030850110149395658513, 14.38405473958341063448838772395
