| L(s) = 1 | − 2.71·2-s − 0.244·3-s + 5.35·4-s − 2.10·5-s + 0.662·6-s − 7-s − 9.10·8-s − 2.94·9-s + 5.71·10-s − 5.79·11-s − 1.30·12-s + 6.38·13-s + 2.71·14-s + 0.514·15-s + 13.9·16-s − 2.07·17-s + 7.97·18-s − 8.19·19-s − 11.2·20-s + 0.244·21-s + 15.7·22-s + 4.78·23-s + 2.22·24-s − 0.564·25-s − 17.3·26-s + 1.44·27-s − 5.35·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 0.140·3-s + 2.67·4-s − 0.941·5-s + 0.270·6-s − 0.377·7-s − 3.21·8-s − 0.980·9-s + 1.80·10-s − 1.74·11-s − 0.377·12-s + 1.77·13-s + 0.724·14-s + 0.132·15-s + 3.49·16-s − 0.502·17-s + 1.87·18-s − 1.88·19-s − 2.52·20-s + 0.0532·21-s + 3.34·22-s + 0.996·23-s + 0.453·24-s − 0.112·25-s − 3.39·26-s + 0.279·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05785004076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05785004076\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 863 | \( 1 - T \) |
| good | 2 | \( 1 + 2.71T + 2T^{2} \) |
| 3 | \( 1 + 0.244T + 3T^{2} \) |
| 5 | \( 1 + 2.10T + 5T^{2} \) |
| 11 | \( 1 + 5.79T + 11T^{2} \) |
| 13 | \( 1 - 6.38T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 + 8.19T + 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 - 4.34T + 29T^{2} \) |
| 31 | \( 1 - 3.75T + 31T^{2} \) |
| 37 | \( 1 + 4.73T + 37T^{2} \) |
| 41 | \( 1 + 7.81T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 7.29T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 5.04T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−8.209856301695092432021211993526, −7.76620167107048938187940094385, −6.81667836009137770999212771304, −6.28491031081764818667356759441, −5.60652555858356892605213035944, −4.33563969330694162392415419750, −3.09770501039980079499567650964, −2.72143525327039602442424346072, −1.50598821091336174033177737518, −0.16817319410203350106775627120,
0.16817319410203350106775627120, 1.50598821091336174033177737518, 2.72143525327039602442424346072, 3.09770501039980079499567650964, 4.33563969330694162392415419750, 5.60652555858356892605213035944, 6.28491031081764818667356759441, 6.81667836009137770999212771304, 7.76620167107048938187940094385, 8.209856301695092432021211993526
