| L(s) = 1 | + 2.07·3-s − 1.28·5-s − 4.21·7-s + 1.28·9-s − 6.44·11-s − 13-s − 2.66·15-s + 7.60·17-s − 4.67·19-s − 8.72·21-s + 6.94·23-s − 3.34·25-s − 3.54·27-s + 29-s − 7.98·31-s − 13.3·33-s + 5.42·35-s + 3.04·37-s − 2.07·39-s + 6.59·41-s + 8.31·43-s − 1.65·45-s − 0.728·47-s + 10.7·49-s + 15.7·51-s + 1.44·53-s + 8.29·55-s + ⋯ |
| L(s) = 1 | + 1.19·3-s − 0.575·5-s − 1.59·7-s + 0.428·9-s − 1.94·11-s − 0.277·13-s − 0.688·15-s + 1.84·17-s − 1.07·19-s − 1.90·21-s + 1.44·23-s − 0.668·25-s − 0.682·27-s + 0.185·29-s − 1.43·31-s − 2.32·33-s + 0.917·35-s + 0.500·37-s − 0.331·39-s + 1.03·41-s + 1.26·43-s − 0.247·45-s − 0.106·47-s + 1.53·49-s + 2.20·51-s + 0.197·53-s + 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.371385786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.371385786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.07T + 3T^{2} \) |
| 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 + 6.44T + 11T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 31 | \( 1 + 7.98T + 31T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 + 0.728T + 47T^{2} \) |
| 53 | \( 1 - 1.44T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 2.99T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 0.514T + 71T^{2} \) |
| 73 | \( 1 + 4.71T + 73T^{2} \) |
| 79 | \( 1 + 5.02T + 79T^{2} \) |
| 83 | \( 1 - 6.53T + 83T^{2} \) |
| 89 | \( 1 - 9.98T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.966915775447031675974969403284, −7.56892266291826427846020831583, −6.95567312450153215089909590232, −5.80130335689771953541693514249, −5.36548590037775607926000994651, −4.10840715468003208252341713742, −3.42246017989939110346133379242, −2.86219561755236263231816885410, −2.31277593350208689455657240446, −0.53872120728769774675575899162,
0.53872120728769774675575899162, 2.31277593350208689455657240446, 2.86219561755236263231816885410, 3.42246017989939110346133379242, 4.10840715468003208252341713742, 5.36548590037775607926000994651, 5.80130335689771953541693514249, 6.95567312450153215089909590232, 7.56892266291826427846020831583, 7.966915775447031675974969403284
