| L(s) = 1 | + 2·2-s + 4·4-s − 1.19·5-s + 8·8-s − 2.39·10-s + 25.7·13-s + 16·16-s + 17.9·17-s − 4.78·20-s − 23.5·25-s + 51.5·26-s − 56.3·29-s + 32·32-s + 35.9·34-s + 55.7·37-s − 9.56·40-s − 80·41-s + 49·49-s − 47.1·50-s + 103.·52-s − 56·53-s − 112.·58-s − 92.9·61-s + 64·64-s − 30.8·65-s + 71.9·68-s + 28.1·73-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 0.239·5-s + 8-s − 0.239·10-s + 1.98·13-s + 16-s + 1.05·17-s − 0.239·20-s − 0.942·25-s + 1.98·26-s − 1.94·29-s + 32-s + 1.05·34-s + 1.50·37-s − 0.239·40-s − 1.95·41-s + 0.999·49-s − 0.942·50-s + 1.98·52-s − 1.05·53-s − 1.94·58-s − 1.52·61-s + 64-s − 0.474·65-s + 1.05·68-s + 0.385·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.225963172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.225963172\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.19T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 25.7T + 169T^{2} \) |
| 17 | \( 1 - 17.9T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 56.3T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 55.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 80T + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 56T + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 92.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 28.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 147.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 130T + 9.40e3T^{2} \) |
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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
−11.42764764534448866915878488045, −10.86483864038721866654262165860, −9.684102258089983676743458883068, −8.324715172802997450492952979663, −7.46367641412922211947498409705, −6.21643905465330138286464466350, −5.53085309479530702136469362043, −4.07502475610214170738817076965, −3.30727797956611518042803844023, −1.54425538553117347082213142157,
1.54425538553117347082213142157, 3.30727797956611518042803844023, 4.07502475610214170738817076965, 5.53085309479530702136469362043, 6.21643905465330138286464466350, 7.46367641412922211947498409705, 8.324715172802997450492952979663, 9.684102258089983676743458883068, 10.86483864038721866654262165860, 11.42764764534448866915878488045
