| L(s) = 1 | + 0.291·2-s − 2.43·3-s − 1.91·4-s + 2.06·5-s − 0.708·6-s − 1.14·8-s + 2.91·9-s + 0.601·10-s − 4.94·11-s + 4.65·12-s − 1.62·13-s − 5.02·15-s + 3.49·16-s + 6.99·17-s + 0.849·18-s − 19-s − 3.95·20-s − 1.44·22-s − 7.41·23-s + 2.77·24-s − 0.732·25-s − 0.473·26-s + 0.206·27-s + 3.88·29-s − 1.46·30-s − 0.808·31-s + 3.30·32-s + ⋯ |
| L(s) = 1 | + 0.206·2-s − 1.40·3-s − 0.957·4-s + 0.923·5-s − 0.289·6-s − 0.403·8-s + 0.971·9-s + 0.190·10-s − 1.49·11-s + 1.34·12-s − 0.450·13-s − 1.29·15-s + 0.874·16-s + 1.69·17-s + 0.200·18-s − 0.229·19-s − 0.884·20-s − 0.307·22-s − 1.54·23-s + 0.566·24-s − 0.146·25-s − 0.0927·26-s + 0.0397·27-s + 0.721·29-s − 0.267·30-s − 0.145·31-s + 0.583·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7700089804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7700089804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.291T + 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 + 1.62T + 13T^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 + 0.808T + 31T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 - 8.50T + 41T^{2} \) |
| 43 | \( 1 - 7.31T + 43T^{2} \) |
| 47 | \( 1 - 5.41T + 47T^{2} \) |
| 53 | \( 1 + 1.99T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 6.51T + 71T^{2} \) |
| 73 | \( 1 - 2.08T + 73T^{2} \) |
| 79 | \( 1 + 2.31T + 79T^{2} \) |
| 83 | \( 1 + 5.01T + 83T^{2} \) |
| 89 | \( 1 + 6.69T + 89T^{2} \) |
| 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
−9.979178096093483593906834826583, −9.734134844669717279524722454379, −8.300661776881813268491982108631, −7.55607071415331699067197850534, −6.13164000396915930037888952998, −5.59215118530479136100922771407, −5.16176195626307861293627252358, −4.06854491297915925016670502671, −2.52099697788367695312034632874, −0.71582433814053977439620672910,
0.71582433814053977439620672910, 2.52099697788367695312034632874, 4.06854491297915925016670502671, 5.16176195626307861293627252358, 5.59215118530479136100922771407, 6.13164000396915930037888952998, 7.55607071415331699067197850534, 8.300661776881813268491982108631, 9.734134844669717279524722454379, 9.979178096093483593906834826583
