| L(s) = 1 | + 0.682·2-s + 2.48·3-s − 1.53·4-s + 5-s + 1.69·6-s + 0.856·7-s − 2.41·8-s + 3.17·9-s + 0.682·10-s + 11-s − 3.80·12-s + 5.50·13-s + 0.585·14-s + 2.48·15-s + 1.41·16-s − 3.50·17-s + 2.16·18-s + 19-s − 1.53·20-s + 2.12·21-s + 0.682·22-s + 6.75·23-s − 5.99·24-s + 25-s + 3.76·26-s + 0.425·27-s − 1.31·28-s + ⋯ |
| L(s) = 1 | + 0.482·2-s + 1.43·3-s − 0.766·4-s + 0.447·5-s + 0.692·6-s + 0.323·7-s − 0.853·8-s + 1.05·9-s + 0.215·10-s + 0.301·11-s − 1.09·12-s + 1.52·13-s + 0.156·14-s + 0.641·15-s + 0.354·16-s − 0.850·17-s + 0.510·18-s + 0.229·19-s − 0.342·20-s + 0.464·21-s + 0.145·22-s + 1.40·23-s − 1.22·24-s + 0.200·25-s + 0.737·26-s + 0.0818·27-s − 0.248·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.146164406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.146164406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.682T + 2T^{2} \) |
| 3 | \( 1 - 2.48T + 3T^{2} \) |
| 7 | \( 1 - 0.856T + 7T^{2} \) |
| 13 | \( 1 - 5.50T + 13T^{2} \) |
| 17 | \( 1 + 3.50T + 17T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 + 0.474T + 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 - 0.420T + 37T^{2} \) |
| 41 | \( 1 - 8.25T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 - 2.48T + 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 3.08T + 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−9.484032003481323963055793432247, −8.994132927050149618027594096704, −8.573397962793327672987637167377, −7.63293175598410780332989690133, −6.47655284972358707911733208397, −5.51345700945337198358912628585, −4.42355400944855068054700303086, −3.63375110434590437091197383835, −2.79548748068295247468293174116, −1.42836725810507927798981883471,
1.42836725810507927798981883471, 2.79548748068295247468293174116, 3.63375110434590437091197383835, 4.42355400944855068054700303086, 5.51345700945337198358912628585, 6.47655284972358707911733208397, 7.63293175598410780332989690133, 8.573397962793327672987637167377, 8.994132927050149618027594096704, 9.484032003481323963055793432247
