| L(s) = 1 | + 0.739·3-s + 1.95·5-s − 2.45·9-s − 4.49·11-s + 5.82·13-s + 1.44·15-s − 1.81·17-s + 3.81·19-s − 2.35·23-s − 1.19·25-s − 4.03·27-s − 5.66·29-s − 9.39·31-s − 3.32·33-s − 8.01·37-s + 4.30·39-s + 12.1·41-s − 9.11·43-s − 4.78·45-s − 3.97·47-s − 1.34·51-s − 5.31·53-s − 8.77·55-s + 2.82·57-s + 3.69·59-s + 7.81·61-s + 11.3·65-s + ⋯ |
| L(s) = 1 | + 0.427·3-s + 0.872·5-s − 0.817·9-s − 1.35·11-s + 1.61·13-s + 0.372·15-s − 0.440·17-s + 0.874·19-s − 0.490·23-s − 0.239·25-s − 0.776·27-s − 1.05·29-s − 1.68·31-s − 0.579·33-s − 1.31·37-s + 0.690·39-s + 1.90·41-s − 1.38·43-s − 0.713·45-s − 0.579·47-s − 0.187·51-s − 0.730·53-s − 1.18·55-s + 0.373·57-s + 0.480·59-s + 1.00·61-s + 1.40·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.739T + 3T^{2} \) |
| 5 | \( 1 - 1.95T + 5T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 19 | \( 1 - 3.81T + 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 + 5.66T + 29T^{2} \) |
| 31 | \( 1 + 9.39T + 31T^{2} \) |
| 37 | \( 1 + 8.01T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 + 5.31T + 53T^{2} \) |
| 59 | \( 1 - 3.69T + 59T^{2} \) |
| 61 | \( 1 - 7.81T + 61T^{2} \) |
| 67 | \( 1 - 1.86T + 67T^{2} \) |
| 71 | \( 1 - 0.149T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 - 7.97T + 97T^{2} \) |
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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
−7.909047067589233961938332234690, −7.19668132159988696920203566800, −6.15314237984889124307677620736, −5.63266402866382799808133242559, −5.19706374686566251647731097808, −3.85459069737060238312949493364, −3.24590938730691307876566950218, −2.31735655130253967587777879622, −1.59665991123123716577188366415, 0,
1.59665991123123716577188366415, 2.31735655130253967587777879622, 3.24590938730691307876566950218, 3.85459069737060238312949493364, 5.19706374686566251647731097808, 5.63266402866382799808133242559, 6.15314237984889124307677620736, 7.19668132159988696920203566800, 7.909047067589233961938332234690
