| L(s) = 1 | + 25·5-s + 132·7-s + 472·11-s − 686·13-s + 1.56e3·17-s + 2.18e3·19-s + 264·23-s + 625·25-s − 170·29-s − 7.27e3·31-s + 3.30e3·35-s − 142·37-s + 1.61e4·41-s + 1.03e4·43-s + 1.85e4·47-s + 617·49-s − 2.15e4·53-s + 1.18e4·55-s + 3.46e4·59-s − 3.57e4·61-s − 1.71e4·65-s + 5.77e3·67-s − 6.90e4·71-s − 7.05e4·73-s + 6.23e4·77-s − 4.76e4·79-s + 7.40e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.01·7-s + 1.17·11-s − 1.12·13-s + 1.31·17-s + 1.38·19-s + 0.104·23-s + 1/5·25-s − 0.0375·29-s − 1.35·31-s + 0.455·35-s − 0.0170·37-s + 1.50·41-s + 0.850·43-s + 1.22·47-s + 0.0367·49-s − 1.05·53-s + 0.525·55-s + 1.29·59-s − 1.22·61-s − 0.503·65-s + 0.157·67-s − 1.62·71-s − 1.54·73-s + 1.19·77-s − 0.858·79-s + 1.17·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.328837066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.328837066\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 - 132 T + p^{5} T^{2} \) |
| 11 | \( 1 - 472 T + p^{5} T^{2} \) |
| 13 | \( 1 + 686 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1562 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2180 T + p^{5} T^{2} \) |
| 23 | \( 1 - 264 T + p^{5} T^{2} \) |
| 29 | \( 1 + 170 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7272 T + p^{5} T^{2} \) |
| 37 | \( 1 + 142 T + p^{5} T^{2} \) |
| 41 | \( 1 - 16198 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10316 T + p^{5} T^{2} \) |
| 47 | \( 1 - 18568 T + p^{5} T^{2} \) |
| 53 | \( 1 + 21514 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34600 T + p^{5} T^{2} \) |
| 61 | \( 1 + 35738 T + p^{5} T^{2} \) |
| 67 | \( 1 - 5772 T + p^{5} T^{2} \) |
| 71 | \( 1 + 69088 T + p^{5} T^{2} \) |
| 73 | \( 1 + 70526 T + p^{5} T^{2} \) |
| 79 | \( 1 + 47640 T + p^{5} T^{2} \) |
| 83 | \( 1 - 74004 T + p^{5} T^{2} \) |
| 89 | \( 1 - 90030 T + p^{5} T^{2} \) |
| 97 | \( 1 + 33502 T + p^{5} T^{2} \) |
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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
−9.525922505192754998411482183569, −9.010525675065401147350314826794, −7.66718537373838444108227842579, −7.31458253549516775718933596578, −5.92278150556469849629766272602, −5.22697051065477244791786687451, −4.23076136364371625357820547276, −3.03078541379242351572686077803, −1.76681026029634059869394578027, −0.921015550957263183538276404528,
0.921015550957263183538276404528, 1.76681026029634059869394578027, 3.03078541379242351572686077803, 4.23076136364371625357820547276, 5.22697051065477244791786687451, 5.92278150556469849629766272602, 7.31458253549516775718933596578, 7.66718537373838444108227842579, 9.010525675065401147350314826794, 9.525922505192754998411482183569
