| L(s) = 1 | − 16·5-s + 12·7-s − 448·11-s − 206·13-s − 1.95e3·17-s + 1.06e3·19-s − 3.71e3·23-s − 2.86e3·25-s − 4.08e3·29-s + 5.32e3·31-s − 192·35-s − 9.69e3·37-s + 9.12e3·41-s + 1.65e4·43-s + 1.42e4·47-s − 1.66e4·49-s + 2.17e4·53-s + 7.16e3·55-s + 3.16e4·59-s − 1.31e4·61-s + 3.29e3·65-s + 2.70e4·67-s − 9.72e3·71-s + 9.04e3·73-s − 5.37e3·77-s − 5.82e4·79-s − 8.63e4·83-s + ⋯ |
| L(s) = 1 | − 0.286·5-s + 0.0925·7-s − 1.11·11-s − 0.338·13-s − 1.63·17-s + 0.676·19-s − 1.46·23-s − 0.918·25-s − 0.900·29-s + 0.995·31-s − 0.0264·35-s − 1.16·37-s + 0.847·41-s + 1.36·43-s + 0.938·47-s − 0.991·49-s + 1.06·53-s + 0.319·55-s + 1.18·59-s − 0.452·61-s + 0.0967·65-s + 0.736·67-s − 0.229·71-s + 0.198·73-s − 0.103·77-s − 1.05·79-s − 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 16 T + p^{5} T^{2} \) |
| 7 | \( 1 - 12 T + p^{5} T^{2} \) |
| 11 | \( 1 + 448 T + p^{5} T^{2} \) |
| 13 | \( 1 + 206 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1952 T + p^{5} T^{2} \) |
| 19 | \( 1 - 56 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 3712 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4080 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5324 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9690 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9120 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16552 T + p^{5} T^{2} \) |
| 47 | \( 1 - 14208 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21776 T + p^{5} T^{2} \) |
| 59 | \( 1 - 31616 T + p^{5} T^{2} \) |
| 61 | \( 1 + 13154 T + p^{5} T^{2} \) |
| 67 | \( 1 - 27056 T + p^{5} T^{2} \) |
| 71 | \( 1 + 9728 T + p^{5} T^{2} \) |
| 73 | \( 1 - 9046 T + p^{5} T^{2} \) |
| 79 | \( 1 + 58292 T + p^{5} T^{2} \) |
| 83 | \( 1 + 86336 T + p^{5} T^{2} \) |
| 89 | \( 1 + 75072 T + p^{5} T^{2} \) |
| 97 | \( 1 - 76046 T + p^{5} T^{2} \) |
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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
−13.15913221709160958092153285870, −11.97193827669132572032215058565, −10.89713242581939046077030530362, −9.738789576309957913496309647296, −8.343612548313516887034408917571, −7.26584283302539812117565008077, −5.67640048418213598169314276280, −4.20448710891994482549871220218, −2.34629939163771345011778760752, 0,
2.34629939163771345011778760752, 4.20448710891994482549871220218, 5.67640048418213598169314276280, 7.26584283302539812117565008077, 8.343612548313516887034408917571, 9.738789576309957913496309647296, 10.89713242581939046077030530362, 11.97193827669132572032215058565, 13.15913221709160958092153285870
