| L(s) = 1 | − 2.46·5-s − 6.46·13-s + 5.92·17-s + 1.07·25-s + 1.53·29-s − 9.39·37-s + 10·41-s − 7·49-s + 14·53-s + 15.3·61-s + 15.9·65-s + 16.8·73-s − 14.6·85-s − 18.8·89-s + 18·97-s − 2·101-s + 14.3·109-s + 20.8·113-s + ⋯ |
| L(s) = 1 | − 1.10·5-s − 1.79·13-s + 1.43·17-s + 0.214·25-s + 0.285·29-s − 1.54·37-s + 1.56·41-s − 49-s + 1.92·53-s + 1.97·61-s + 1.97·65-s + 1.97·73-s − 1.58·85-s − 1.99·89-s + 1.82·97-s − 0.199·101-s + 1.37·109-s + 1.96·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.126942554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.126942554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 18.8T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.767238603902416120145703922562, −7.963803314166007795899223243328, −7.44527110345090499550764015477, −6.85642536690910951543973029033, −5.60665368717929103243430413214, −4.95626535723846842116687670342, −4.04048700673600658805380599089, −3.25183516051767063993166898578, −2.22899656870296976889201598949, −0.65084565961050136195540735714,
0.65084565961050136195540735714, 2.22899656870296976889201598949, 3.25183516051767063993166898578, 4.04048700673600658805380599089, 4.95626535723846842116687670342, 5.60665368717929103243430413214, 6.85642536690910951543973029033, 7.44527110345090499550764015477, 7.963803314166007795899223243328, 8.767238603902416120145703922562
