| L(s) = 1 | − 3·3-s − 16·5-s + 8·7-s + 9·9-s + 38·11-s − 13·13-s + 48·15-s − 78·17-s + 72·19-s − 24·21-s + 52·23-s + 131·25-s − 27·27-s + 242·29-s − 76·31-s − 114·33-s − 128·35-s + 342·37-s + 39·39-s − 336·41-s − 76·43-s − 144·45-s − 94·47-s − 279·49-s + 234·51-s − 450·53-s − 608·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.43·5-s + 0.431·7-s + 1/3·9-s + 1.04·11-s − 0.277·13-s + 0.826·15-s − 1.11·17-s + 0.869·19-s − 0.249·21-s + 0.471·23-s + 1.04·25-s − 0.192·27-s + 1.54·29-s − 0.440·31-s − 0.601·33-s − 0.618·35-s + 1.51·37-s + 0.160·39-s − 1.27·41-s − 0.269·43-s − 0.477·45-s − 0.291·47-s − 0.813·49-s + 0.642·51-s − 1.16·53-s − 1.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 16 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 72 T + p^{3} T^{2} \) |
| 23 | \( 1 - 52 T + p^{3} T^{2} \) |
| 29 | \( 1 - 242 T + p^{3} T^{2} \) |
| 31 | \( 1 + 76 T + p^{3} T^{2} \) |
| 37 | \( 1 - 342 T + p^{3} T^{2} \) |
| 41 | \( 1 + 336 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 53 | \( 1 + 450 T + p^{3} T^{2} \) |
| 59 | \( 1 + 854 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 - 908 T + p^{3} T^{2} \) |
| 71 | \( 1 + 838 T + p^{3} T^{2} \) |
| 73 | \( 1 + 970 T + p^{3} T^{2} \) |
| 79 | \( 1 - 352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 474 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1452 T + p^{3} T^{2} \) |
| 97 | \( 1 + 562 T + p^{3} T^{2} \) |
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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
−9.799638809905353358365538683543, −8.801480300431704813714619564420, −7.949432028295995330343858165553, −7.09173081727496385400962160387, −6.31450630548210047107710648764, −4.85246294091664832824178532346, −4.29913056238721882678960563529, −3.13448336692137219508657297049, −1.30849712099864171421914101475, 0,
1.30849712099864171421914101475, 3.13448336692137219508657297049, 4.29913056238721882678960563529, 4.85246294091664832824178532346, 6.31450630548210047107710648764, 7.09173081727496385400962160387, 7.949432028295995330343858165553, 8.801480300431704813714619564420, 9.799638809905353358365538683543
