| L(s) = 1 | − 10.0·2-s + 68.1·4-s − 70.3·5-s − 362.·8-s + 704.·10-s + 731.·11-s + 899.·13-s + 1.44e3·16-s + 1.39e3·17-s − 190.·19-s − 4.79e3·20-s − 7.32e3·22-s − 42.9·23-s + 1.82e3·25-s − 9.00e3·26-s + 7.74e3·29-s + 1.17e3·31-s − 2.85e3·32-s − 1.39e4·34-s + 9.28e3·37-s + 1.90e3·38-s + 2.54e4·40-s − 1.34e4·41-s + 6.03e3·43-s + 4.98e4·44-s + 430.·46-s + 3.24e3·47-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.13·4-s − 1.25·5-s − 2.00·8-s + 2.22·10-s + 1.82·11-s + 1.47·13-s + 1.40·16-s + 1.16·17-s − 0.120·19-s − 2.68·20-s − 3.22·22-s − 0.0169·23-s + 0.583·25-s − 2.61·26-s + 1.71·29-s + 0.220·31-s − 0.492·32-s − 2.06·34-s + 1.11·37-s + 0.213·38-s + 2.51·40-s − 1.24·41-s + 0.497·43-s + 3.88·44-s + 0.0299·46-s + 0.214·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9323818838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9323818838\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 10.0T + 32T^{2} \) |
| 5 | \( 1 + 70.3T + 3.12e3T^{2} \) |
| 11 | \( 1 - 731.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 899.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.39e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 190.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 42.9T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.74e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.28e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.03e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.24e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.43e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.68e4T + 8.58e9T^{2} \) |
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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
−10.18025393270499152350602795214, −9.250582103705907594734114610991, −8.453172087300750684657673561489, −7.953616993154401340364797231695, −6.87652977902913361024642602576, −6.19048464062833239169087432654, −4.17611291045419306133614161158, −3.21519642319761477829202733349, −1.39031295884614459618954061233, −0.75530530917433973799639384177,
0.75530530917433973799639384177, 1.39031295884614459618954061233, 3.21519642319761477829202733349, 4.17611291045419306133614161158, 6.19048464062833239169087432654, 6.87652977902913361024642602576, 7.953616993154401340364797231695, 8.453172087300750684657673561489, 9.250582103705907594734114610991, 10.18025393270499152350602795214
