| L(s) = 1 | + 3.82·2-s + 9·3-s − 17.4·4-s − 16.4·5-s + 34.3·6-s + 229.·7-s − 188.·8-s + 81·9-s − 62.9·10-s + 504.·11-s − 156.·12-s − 879.·13-s + 876.·14-s − 148.·15-s − 164.·16-s − 1.09e3·17-s + 309.·18-s + 2.79e3·19-s + 286.·20-s + 2.06e3·21-s + 1.92e3·22-s + 2.91e3·23-s − 1.69e3·24-s − 2.85e3·25-s − 3.36e3·26-s + 729·27-s − 3.99e3·28-s + ⋯ |
| L(s) = 1 | + 0.675·2-s + 0.577·3-s − 0.543·4-s − 0.294·5-s + 0.389·6-s + 1.76·7-s − 1.04·8-s + 0.333·9-s − 0.198·10-s + 1.25·11-s − 0.313·12-s − 1.44·13-s + 1.19·14-s − 0.170·15-s − 0.160·16-s − 0.916·17-s + 0.225·18-s + 1.77·19-s + 0.160·20-s + 1.02·21-s + 0.848·22-s + 1.14·23-s − 0.602·24-s − 0.913·25-s − 0.975·26-s + 0.192·27-s − 0.961·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.620756501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.620756501\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 103 | \( 1 + 1.06e4T \) |
| good | 2 | \( 1 - 3.82T + 32T^{2} \) |
| 5 | \( 1 + 16.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 229.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 504.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 879.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.91e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.95e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.86e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.96e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.62e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.76e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.22e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.88e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.33e4T + 8.58e9T^{2} \) |
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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
−11.26293731763774611309392670039, −9.574913119944782260493671785710, −9.061890204239595178248085104962, −7.937273644384225118227426265053, −7.18412062954343213356295532469, −5.49601961378933959897961111858, −4.64215248330202177674795601342, −3.91260008790958334813893256157, −2.44746660396284800195248280589, −1.00593223922880607812731409164,
1.00593223922880607812731409164, 2.44746660396284800195248280589, 3.91260008790958334813893256157, 4.64215248330202177674795601342, 5.49601961378933959897961111858, 7.18412062954343213356295532469, 7.937273644384225118227426265053, 9.061890204239595178248085104962, 9.574913119944782260493671785710, 11.26293731763774611309392670039
