| L(s) = 1 | − 1.59·5-s − 4.94·7-s + 5.59·11-s − 3.49·13-s − 7.09·17-s + 19-s − 3.19·23-s − 2.44·25-s + 4.69·29-s + 6·31-s + 7.89·35-s + 4·37-s − 2.50·41-s + 8.94·43-s − 6.28·47-s + 17.4·49-s − 5.69·53-s − 8.94·55-s − 6.99·59-s − 3.44·61-s + 5.58·65-s + 13.3·67-s + 13.3·71-s − 2.44·73-s − 27.6·77-s + 9.49·79-s − 0.691·83-s + ⋯ |
| L(s) = 1 | − 0.714·5-s − 1.86·7-s + 1.68·11-s − 0.969·13-s − 1.72·17-s + 0.229·19-s − 0.666·23-s − 0.489·25-s + 0.871·29-s + 1.07·31-s + 1.33·35-s + 0.657·37-s − 0.390·41-s + 1.36·43-s − 0.917·47-s + 2.49·49-s − 0.782·53-s − 1.20·55-s − 0.910·59-s − 0.441·61-s + 0.692·65-s + 1.63·67-s + 1.58·71-s − 0.285·73-s − 3.15·77-s + 1.06·79-s − 0.0759·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8779035100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8779035100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 + 4.94T + 7T^{2} \) |
| 11 | \( 1 - 5.59T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 + 7.09T + 17T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 - 4.69T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 6.28T + 47T^{2} \) |
| 53 | \( 1 + 5.69T + 53T^{2} \) |
| 59 | \( 1 + 6.99T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 2.44T + 73T^{2} \) |
| 79 | \( 1 - 9.49T + 79T^{2} \) |
| 83 | \( 1 + 0.691T + 83T^{2} \) |
| 89 | \( 1 + 8.69T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−9.008323962676605497206953300904, −8.082143895046820758007360732650, −7.10355983907956517634003002087, −6.50493518364678541419156184543, −6.15057102374048773928560667364, −4.63299619285506807206581955297, −4.02573810876410922499598196905, −3.24617619502029441718846276838, −2.26284109138273016309241622894, −0.55585790781882677070066554392,
0.55585790781882677070066554392, 2.26284109138273016309241622894, 3.24617619502029441718846276838, 4.02573810876410922499598196905, 4.63299619285506807206581955297, 6.15057102374048773928560667364, 6.50493518364678541419156184543, 7.10355983907956517634003002087, 8.082143895046820758007360732650, 9.008323962676605497206953300904
