| L(s) = 1 | + 5-s + 9·7-s + 17·11-s − 44·13-s + 56·17-s + 94·19-s + 50·23-s − 124·25-s − 30·29-s + 139·31-s + 9·35-s − 174·37-s + 318·41-s + 242·43-s + 630·47-s − 262·49-s + 547·53-s + 17·55-s + 236·59-s + 328·61-s − 44·65-s − 614·67-s − 296·71-s + 433·73-s + 153·77-s + 56·79-s + 1.22e3·83-s + ⋯ |
| L(s) = 1 | + 0.0894·5-s + 0.485·7-s + 0.465·11-s − 0.938·13-s + 0.798·17-s + 1.13·19-s + 0.453·23-s − 0.991·25-s − 0.192·29-s + 0.805·31-s + 0.0434·35-s − 0.773·37-s + 1.21·41-s + 0.858·43-s + 1.95·47-s − 0.763·49-s + 1.41·53-s + 0.0416·55-s + 0.520·59-s + 0.688·61-s − 0.0839·65-s − 1.11·67-s − 0.494·71-s + 0.694·73-s + 0.226·77-s + 0.0797·79-s + 1.62·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.125636817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.125636817\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 17 T + p^{3} T^{2} \) |
| 13 | \( 1 + 44 T + p^{3} T^{2} \) |
| 17 | \( 1 - 56 T + p^{3} T^{2} \) |
| 19 | \( 1 - 94 T + p^{3} T^{2} \) |
| 23 | \( 1 - 50 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 139 T + p^{3} T^{2} \) |
| 37 | \( 1 + 174 T + p^{3} T^{2} \) |
| 41 | \( 1 - 318 T + p^{3} T^{2} \) |
| 43 | \( 1 - 242 T + p^{3} T^{2} \) |
| 47 | \( 1 - 630 T + p^{3} T^{2} \) |
| 53 | \( 1 - 547 T + p^{3} T^{2} \) |
| 59 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 61 | \( 1 - 328 T + p^{3} T^{2} \) |
| 67 | \( 1 + 614 T + p^{3} T^{2} \) |
| 71 | \( 1 + 296 T + p^{3} T^{2} \) |
| 73 | \( 1 - 433 T + p^{3} T^{2} \) |
| 79 | \( 1 - 56 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1225 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1506 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1391 T + p^{3} T^{2} \) |
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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
−10.69845943465627517512654514654, −9.789197452276592330133399227852, −9.047443755980716455768347628346, −7.82536776899000508105750824032, −7.21143185673041049443760005927, −5.87763965558438420785137122933, −5.00102733274800659427120974222, −3.80219943642894295318344822055, −2.44529013802392391127208190595, −0.981512874767833974152911838547,
0.981512874767833974152911838547, 2.44529013802392391127208190595, 3.80219943642894295318344822055, 5.00102733274800659427120974222, 5.87763965558438420785137122933, 7.21143185673041049443760005927, 7.82536776899000508105750824032, 9.047443755980716455768347628346, 9.789197452276592330133399227852, 10.69845943465627517512654514654
