| L(s) = 1 | + 9·3-s − 25·5-s − 244·7-s + 81·9-s − 144·11-s + 50·13-s − 225·15-s − 1.91e3·17-s + 140·19-s − 2.19e3·21-s − 624·23-s + 625·25-s + 729·27-s − 3.12e3·29-s − 5.17e3·31-s − 1.29e3·33-s + 6.10e3·35-s + 1.56e4·37-s + 450·39-s + 1.25e4·41-s + 1.15e4·43-s − 2.02e3·45-s − 2.67e4·47-s + 4.27e4·49-s − 1.72e4·51-s − 1.91e4·53-s + 3.60e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.88·7-s + 1/3·9-s − 0.358·11-s + 0.0820·13-s − 0.258·15-s − 1.60·17-s + 0.0889·19-s − 1.08·21-s − 0.245·23-s + 1/5·25-s + 0.192·27-s − 0.690·29-s − 0.967·31-s − 0.207·33-s + 0.841·35-s + 1.88·37-s + 0.0473·39-s + 1.16·41-s + 0.949·43-s − 0.149·45-s − 1.76·47-s + 2.54·49-s − 0.927·51-s − 0.936·53-s + 0.160·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 244 T + p^{5} T^{2} \) |
| 11 | \( 1 + 144 T + p^{5} T^{2} \) |
| 13 | \( 1 - 50 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1914 T + p^{5} T^{2} \) |
| 19 | \( 1 - 140 T + p^{5} T^{2} \) |
| 23 | \( 1 + 624 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3126 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5176 T + p^{5} T^{2} \) |
| 37 | \( 1 - 15698 T + p^{5} T^{2} \) |
| 41 | \( 1 - 12570 T + p^{5} T^{2} \) |
| 43 | \( 1 - 11516 T + p^{5} T^{2} \) |
| 47 | \( 1 + 26736 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19158 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27984 T + p^{5} T^{2} \) |
| 61 | \( 1 - 22022 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12676 T + p^{5} T^{2} \) |
| 71 | \( 1 + 59520 T + p^{5} T^{2} \) |
| 73 | \( 1 + 67102 T + p^{5} T^{2} \) |
| 79 | \( 1 - 11048 T + p^{5} T^{2} \) |
| 83 | \( 1 + 115284 T + p^{5} T^{2} \) |
| 89 | \( 1 - 73650 T + p^{5} T^{2} \) |
| 97 | \( 1 - 35522 T + p^{5} T^{2} \) |
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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
−13.19608897158766904744572618084, −12.84904397183913869133922437415, −11.18933965496497813090757758486, −9.811871825171579612193623302843, −8.921632964727822190126097221417, −7.39339626679652731458093083983, −6.20538435357006367255726696052, −4.05019212250535946607415776013, −2.72264141700998244990770226159, 0,
2.72264141700998244990770226159, 4.05019212250535946607415776013, 6.20538435357006367255726696052, 7.39339626679652731458093083983, 8.921632964727822190126097221417, 9.811871825171579612193623302843, 11.18933965496497813090757758486, 12.84904397183913869133922437415, 13.19608897158766904744572618084
