| L(s) = 1 | + 2.29·5-s − 7-s − 5.01·11-s − 4.55·13-s + 0.762·17-s + 3.01·19-s + 1.53·23-s + 0.278·25-s + 3.29·29-s − 3.29·31-s − 2.29·35-s + 4.72·37-s + 0.297·41-s + 3.83·43-s − 5.79·47-s + 49-s + 11.3·53-s − 11.5·55-s + 3.59·59-s + 2.42·61-s − 10.4·65-s − 4.12·67-s + 2.46·71-s + 16.6·73-s + 5.01·77-s + 5.36·79-s − 1.27·83-s + ⋯ |
| L(s) = 1 | + 1.02·5-s − 0.377·7-s − 1.51·11-s − 1.26·13-s + 0.185·17-s + 0.692·19-s + 0.319·23-s + 0.0557·25-s + 0.612·29-s − 0.592·31-s − 0.388·35-s + 0.776·37-s + 0.0464·41-s + 0.584·43-s − 0.844·47-s + 0.142·49-s + 1.56·53-s − 1.55·55-s + 0.468·59-s + 0.310·61-s − 1.29·65-s − 0.504·67-s + 0.292·71-s + 1.94·73-s + 0.571·77-s + 0.603·79-s − 0.140·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.804422926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.804422926\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2.29T + 5T^{2} \) |
| 11 | \( 1 + 5.01T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 - 0.762T + 17T^{2} \) |
| 19 | \( 1 - 3.01T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 - 3.29T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 - 4.72T + 37T^{2} \) |
| 41 | \( 1 - 0.297T + 41T^{2} \) |
| 43 | \( 1 - 3.83T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 3.59T + 59T^{2} \) |
| 61 | \( 1 - 2.42T + 61T^{2} \) |
| 67 | \( 1 + 4.12T + 67T^{2} \) |
| 71 | \( 1 - 2.46T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 5.36T + 79T^{2} \) |
| 83 | \( 1 + 1.27T + 83T^{2} \) |
| 89 | \( 1 - 9.57T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.898506186254039527033135694817, −7.44808251368860551437956573256, −6.65275006126495549150240383142, −5.80057510598067748697111354935, −5.27026939780687206005627221816, −4.71050144729925988210262325140, −3.46193659640465037868618973889, −2.58595649955348416239391220494, −2.12507417857292595973522111197, −0.67621378388326198620621996348,
0.67621378388326198620621996348, 2.12507417857292595973522111197, 2.58595649955348416239391220494, 3.46193659640465037868618973889, 4.71050144729925988210262325140, 5.27026939780687206005627221816, 5.80057510598067748697111354935, 6.65275006126495549150240383142, 7.44808251368860551437956573256, 7.898506186254039527033135694817
