| L(s) = 1 | − 0.732·3-s + 5-s + 1.26·7-s − 2.46·9-s + 3.46·11-s − 3.46·13-s − 0.732·15-s + 3.46·17-s − 2·19-s − 0.928·21-s + 8.19·23-s + 25-s + 4·27-s + 9.46·31-s − 2.53·33-s + 1.26·35-s − 6·37-s + 2.53·39-s − 2.53·41-s − 10.1·43-s − 2.46·45-s + 8.19·47-s − 5.39·49-s − 2.53·51-s + 10.3·53-s + 3.46·55-s + 1.46·57-s + ⋯ |
| L(s) = 1 | − 0.422·3-s + 0.447·5-s + 0.479·7-s − 0.821·9-s + 1.04·11-s − 0.960·13-s − 0.189·15-s + 0.840·17-s − 0.458·19-s − 0.202·21-s + 1.70·23-s + 0.200·25-s + 0.769·27-s + 1.69·31-s − 0.441·33-s + 0.214·35-s − 0.986·37-s + 0.406·39-s − 0.396·41-s − 1.55·43-s − 0.367·45-s + 1.19·47-s − 0.770·49-s − 0.355·51-s + 1.42·53-s + 0.467·55-s + 0.193·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.607776162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.607776162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.709365929211758930633779755935, −8.837846762796221502353727919303, −8.190125431894169279004739824631, −6.99883669582351890845569233218, −6.42542475179588501980946204133, −5.32115649095844684880089566713, −4.84973073848680922881631775701, −3.48293374064653797372379883625, −2.38192986440091497987991863021, −0.995301817899201935474387564944,
0.995301817899201935474387564944, 2.38192986440091497987991863021, 3.48293374064653797372379883625, 4.84973073848680922881631775701, 5.32115649095844684880089566713, 6.42542475179588501980946204133, 6.99883669582351890845569233218, 8.190125431894169279004739824631, 8.837846762796221502353727919303, 9.709365929211758930633779755935
