| L(s) = 1 | − 0.144·2-s − 9.93·3-s − 31.9·4-s + 1.43·6-s − 198.·7-s + 9.22·8-s − 144.·9-s + 116.·11-s + 317.·12-s + 353.·13-s + 28.6·14-s + 1.02e3·16-s − 1.70e3·17-s + 20.8·18-s − 2.48e3·19-s + 1.97e3·21-s − 16.7·22-s + 727.·23-s − 91.6·24-s − 50.9·26-s + 3.84e3·27-s + 6.36e3·28-s + 829.·29-s − 1.22e3·31-s − 442.·32-s − 1.15e3·33-s + 245.·34-s + ⋯ |
| L(s) = 1 | − 0.0254·2-s − 0.637·3-s − 0.999·4-s + 0.0162·6-s − 1.53·7-s + 0.0509·8-s − 0.593·9-s + 0.289·11-s + 0.636·12-s + 0.580·13-s + 0.0391·14-s + 0.998·16-s − 1.43·17-s + 0.0151·18-s − 1.58·19-s + 0.978·21-s − 0.00737·22-s + 0.286·23-s − 0.0324·24-s − 0.0147·26-s + 1.01·27-s + 1.53·28-s + 0.183·29-s − 0.229·31-s − 0.0763·32-s − 0.184·33-s + 0.0364·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 0.144T + 32T^{2} \) |
| 3 | \( 1 + 9.93T + 243T^{2} \) |
| 7 | \( 1 + 198.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 116.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 353.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.70e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.48e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 727.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 829.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.22e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.53e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 5.09e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.21e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.10e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.23e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.02e5T + 8.58e9T^{2} \) |
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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
−8.972002941675201388852810360089, −8.147263820630908285155381454061, −6.61158857731317804106543755802, −6.35526844044479109035720521248, −5.37387541871498466191482548226, −4.32338579235059884167998209278, −3.58934663329436431030280227463, −2.42513887552860415249905118273, −0.70195229506438240092762656031, 0,
0.70195229506438240092762656031, 2.42513887552860415249905118273, 3.58934663329436431030280227463, 4.32338579235059884167998209278, 5.37387541871498466191482548226, 6.35526844044479109035720521248, 6.61158857731317804106543755802, 8.147263820630908285155381454061, 8.972002941675201388852810360089
