| L(s) = 1 | + 38.9·2-s − 533.·4-s − 3.38e3·5-s + 3.20e4·7-s − 1.00e5·8-s − 1.31e5·10-s + 6.72e5·11-s − 1.69e6·13-s + 1.24e6·14-s − 2.81e6·16-s − 5.41e6·17-s + 8.84e6·19-s + 1.80e6·20-s + 2.61e7·22-s − 5.22e6·23-s − 3.73e7·25-s − 6.60e7·26-s − 1.70e7·28-s + 7.23e7·29-s + 2.06e8·31-s + 9.60e7·32-s − 2.10e8·34-s − 1.08e8·35-s + 7.22e8·37-s + 3.44e8·38-s + 3.39e8·40-s + 4.29e8·41-s + ⋯ |
| L(s) = 1 | + 0.860·2-s − 0.260·4-s − 0.484·5-s + 0.720·7-s − 1.08·8-s − 0.416·10-s + 1.25·11-s − 1.26·13-s + 0.619·14-s − 0.671·16-s − 0.924·17-s + 0.819·19-s + 0.126·20-s + 1.08·22-s − 0.169·23-s − 0.765·25-s − 1.08·26-s − 0.187·28-s + 0.654·29-s + 1.29·31-s + 0.506·32-s − 0.795·34-s − 0.348·35-s + 1.71·37-s + 0.704·38-s + 0.524·40-s + 0.578·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.567836457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.567836457\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 38.9T + 2.04e3T^{2} \) |
| 5 | \( 1 + 3.38e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.20e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 6.72e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.69e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.41e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.84e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.22e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 7.23e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.06e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 7.22e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 4.29e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.37e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 7.40e7T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.54e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 3.61e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 6.16e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 6.23e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.73e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.89e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.58e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.49e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.94e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 6.65e10T + 7.15e21T^{2} \) |
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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
−11.95799017895248668420108178365, −11.60364389954845948073482330027, −9.811156245048089334645963501145, −8.763241415836236894946824763392, −7.47265649583794369656941162950, −6.09229979962573886306475397927, −4.71441840736527657237252584897, −4.07059021525319392863725063273, −2.54723293853977815039224814343, −0.76273872210096868901037398378,
0.76273872210096868901037398378, 2.54723293853977815039224814343, 4.07059021525319392863725063273, 4.71441840736527657237252584897, 6.09229979962573886306475397927, 7.47265649583794369656941162950, 8.763241415836236894946824763392, 9.811156245048089334645963501145, 11.60364389954845948073482330027, 11.95799017895248668420108178365
