| L(s) = 1 | + 9·3-s + 69.4·5-s − 8.69·7-s + 81·9-s − 121·11-s − 970.·13-s + 625.·15-s − 424.·17-s + 1.43e3·19-s − 78.2·21-s − 2.85e3·23-s + 1.69e3·25-s + 729·27-s − 7.46e3·29-s − 1.03e4·31-s − 1.08e3·33-s − 603.·35-s + 167.·37-s − 8.73e3·39-s + 5.68e3·41-s − 2.11e4·43-s + 5.62e3·45-s + 9.78e3·47-s − 1.67e4·49-s − 3.82e3·51-s + 2.56e4·53-s − 8.40e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.24·5-s − 0.0670·7-s + 0.333·9-s − 0.301·11-s − 1.59·13-s + 0.717·15-s − 0.356·17-s + 0.909·19-s − 0.0387·21-s − 1.12·23-s + 0.543·25-s + 0.192·27-s − 1.64·29-s − 1.93·31-s − 0.174·33-s − 0.0833·35-s + 0.0200·37-s − 0.919·39-s + 0.527·41-s − 1.74·43-s + 0.414·45-s + 0.646·47-s − 0.995·49-s − 0.205·51-s + 1.25·53-s − 0.374·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 11 | \( 1 + 121T \) |
| good | 5 | \( 1 - 69.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 8.69T + 1.68e4T^{2} \) |
| 13 | \( 1 + 970.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 424.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 167.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.68e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.78e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.28e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.31e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.18e4T + 8.58e9T^{2} \) |
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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
−9.703215572062441317245393192169, −8.965979035562144122186727757937, −7.71868700221813782379197477326, −7.04101858461517513295100667821, −5.76209606533280730462095125864, −5.08135487647118570739368664777, −3.68737250931299040823567454429, −2.40794943808158325867742960526, −1.78946640534122851517183591962, 0,
1.78946640534122851517183591962, 2.40794943808158325867742960526, 3.68737250931299040823567454429, 5.08135487647118570739368664777, 5.76209606533280730462095125864, 7.04101858461517513295100667821, 7.71868700221813782379197477326, 8.965979035562144122186727757937, 9.703215572062441317245393192169
