| L(s) = 1 | − 3-s − 7-s + 9-s − 3.26·11-s − 0.340·13-s − 5.75·17-s − 6.49·19-s + 21-s + 8.49·23-s − 27-s − 2·29-s − 8.34·31-s + 3.26·33-s + 6.15·37-s + 0.340·39-s + 0.340·41-s + 8.68·43-s + 49-s + 5.75·51-s + 8.34·53-s + 6.49·57-s + 6.83·59-s + 15.3·61-s − 63-s − 14.8·67-s − 8.49·69-s − 15.9·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 0.333·9-s − 0.983·11-s − 0.0943·13-s − 1.39·17-s − 1.49·19-s + 0.218·21-s + 1.77·23-s − 0.192·27-s − 0.371·29-s − 1.49·31-s + 0.567·33-s + 1.01·37-s + 0.0544·39-s + 0.0531·41-s + 1.32·43-s + 0.142·49-s + 0.806·51-s + 1.14·53-s + 0.860·57-s + 0.890·59-s + 1.96·61-s − 0.125·63-s − 1.81·67-s − 1.02·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8882354686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8882354686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 + 0.340T + 13T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 - 8.49T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8.34T + 31T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 - 0.340T + 41T^{2} \) |
| 43 | \( 1 - 8.68T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.34T + 53T^{2} \) |
| 59 | \( 1 - 6.83T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 6.49T + 97T^{2} \) |
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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
−8.613486357165661954550048906599, −7.39742336360968083806906143127, −7.03694358958118023693915619967, −6.14392777364396821979787678204, −5.50721772760819694669428160267, −4.65782248287610845026908456887, −4.00865267580996968367031008428, −2.80203674624821573858188416216, −2.04914929174397113148928453626, −0.52470202096545724293687002258,
0.52470202096545724293687002258, 2.04914929174397113148928453626, 2.80203674624821573858188416216, 4.00865267580996968367031008428, 4.65782248287610845026908456887, 5.50721772760819694669428160267, 6.14392777364396821979787678204, 7.03694358958118023693915619967, 7.39742336360968083806906143127, 8.613486357165661954550048906599
