| L(s) = 1 | − 1.23·3-s + 2·5-s − 7-s − 1.47·9-s − 11-s − 3.23·13-s − 2.47·15-s − 3.23·17-s + 6.47·19-s + 1.23·21-s − 2.47·23-s − 25-s + 5.52·27-s − 8.47·29-s + 2.76·31-s + 1.23·33-s − 2·35-s + 8.47·37-s + 4.00·39-s − 11.2·41-s + 8·43-s − 2.94·45-s − 2.76·47-s + 49-s + 4.00·51-s + 0.472·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.713·3-s + 0.894·5-s − 0.377·7-s − 0.490·9-s − 0.301·11-s − 0.897·13-s − 0.638·15-s − 0.784·17-s + 1.48·19-s + 0.269·21-s − 0.515·23-s − 0.200·25-s + 1.06·27-s − 1.57·29-s + 0.496·31-s + 0.215·33-s − 0.338·35-s + 1.39·37-s + 0.640·39-s − 1.75·41-s + 1.21·43-s − 0.438·45-s − 0.403·47-s + 0.142·49-s + 0.560·51-s + 0.0648·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143173021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143173021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 0.763T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.204056237718739030083485226928, −7.44254596192400271370264183793, −6.67265871685648006506346840510, −5.98129804351336996481456739541, −5.41564164386509901765585986166, −4.87571042961561175940649671940, −3.72327973263700053710578194480, −2.71358764247186889692001211993, −1.99454974355001730051605564834, −0.58302407435466203941936655036,
0.58302407435466203941936655036, 1.99454974355001730051605564834, 2.71358764247186889692001211993, 3.72327973263700053710578194480, 4.87571042961561175940649671940, 5.41564164386509901765585986166, 5.98129804351336996481456739541, 6.67265871685648006506346840510, 7.44254596192400271370264183793, 8.204056237718739030083485226928
