| L(s) = 1 | − 2.61·2-s − 3.37·3-s + 4.85·4-s + 3.62·5-s + 8.82·6-s − 2.17·7-s − 7.47·8-s + 8.36·9-s − 9.50·10-s − 5.68·11-s − 16.3·12-s − 3.89·13-s + 5.69·14-s − 12.2·15-s + 9.85·16-s − 4.90·17-s − 21.9·18-s + 3.00·19-s + 17.6·20-s + 7.33·21-s + 14.8·22-s + 1.00·23-s + 25.1·24-s + 8.16·25-s + 10.1·26-s − 18.0·27-s − 10.5·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.94·3-s + 2.42·4-s + 1.62·5-s + 3.60·6-s − 0.822·7-s − 2.64·8-s + 2.78·9-s − 3.00·10-s − 1.71·11-s − 4.72·12-s − 1.07·13-s + 1.52·14-s − 3.15·15-s + 2.46·16-s − 1.18·17-s − 5.16·18-s + 0.689·19-s + 3.93·20-s + 1.60·21-s + 3.17·22-s + 0.209·23-s + 5.14·24-s + 1.63·25-s + 1.99·26-s − 3.48·27-s − 1.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05470841818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05470841818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 3.37T + 3T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 5.68T + 11T^{2} \) |
| 13 | \( 1 + 3.89T + 13T^{2} \) |
| 17 | \( 1 + 4.90T + 17T^{2} \) |
| 19 | \( 1 - 3.00T + 19T^{2} \) |
| 23 | \( 1 - 1.00T + 23T^{2} \) |
| 29 | \( 1 - 3.18T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 + 9.30T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 0.229T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 + 9.35T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 + 4.22T + 67T^{2} \) |
| 71 | \( 1 + 4.82T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 1.48T + 83T^{2} \) |
| 89 | \( 1 - 0.582T + 89T^{2} \) |
| 97 | \( 1 + 8.11T + 97T^{2} \) |
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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
−7.56442591039018354029099767331, −7.14616577653949971209999860972, −6.45980315236624865193386436192, −6.10179254176610284611183685493, −5.22120295460733496446997133589, −4.93883387016911704395239434119, −2.98788968823573616392328219145, −2.13970275984314054335025524321, −1.46518572171143568869514681235, −0.17005550359480388410404930722,
0.17005550359480388410404930722, 1.46518572171143568869514681235, 2.13970275984314054335025524321, 2.98788968823573616392328219145, 4.93883387016911704395239434119, 5.22120295460733496446997133589, 6.10179254176610284611183685493, 6.45980315236624865193386436192, 7.14616577653949971209999860972, 7.56442591039018354029099767331
