| L(s) = 1 | + 2.37·2-s + 2.26·3-s + 3.63·4-s − 2.08·5-s + 5.36·6-s + 7-s + 3.88·8-s + 2.11·9-s − 4.94·10-s + 11-s + 8.22·12-s + 13-s + 2.37·14-s − 4.70·15-s + 1.95·16-s + 3.85·17-s + 5.01·18-s + 1.83·19-s − 7.56·20-s + 2.26·21-s + 2.37·22-s − 4.59·23-s + 8.78·24-s − 0.669·25-s + 2.37·26-s − 2.00·27-s + 3.63·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 1.30·3-s + 1.81·4-s − 0.930·5-s + 2.19·6-s + 0.377·7-s + 1.37·8-s + 0.704·9-s − 1.56·10-s + 0.301·11-s + 2.37·12-s + 0.277·13-s + 0.634·14-s − 1.21·15-s + 0.487·16-s + 0.935·17-s + 1.18·18-s + 0.421·19-s − 1.69·20-s + 0.493·21-s + 0.506·22-s − 0.958·23-s + 1.79·24-s − 0.133·25-s + 0.465·26-s − 0.386·27-s + 0.687·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.523022467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.523022467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 3 | \( 1 - 2.26T + 3T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 1.83T + 19T^{2} \) |
| 23 | \( 1 + 4.59T + 23T^{2} \) |
| 29 | \( 1 - 0.128T + 29T^{2} \) |
| 31 | \( 1 - 3.57T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 9.10T + 59T^{2} \) |
| 61 | \( 1 - 4.91T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 4.81T + 73T^{2} \) |
| 79 | \( 1 - 4.76T + 79T^{2} \) |
| 83 | \( 1 + 4.99T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 6.93T + 97T^{2} \) |
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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
−10.04032203908265078567178897451, −8.955166620761583546076262497329, −8.024148810661537535199895714232, −7.55673433754708634935310150831, −6.45877964754133375015080142840, −5.42089100528302861214854850287, −4.39499556117707770816623850890, −3.63460253654760430893106204583, −3.10745439784079463266932534955, −1.87276591804180359206749970813,
1.87276591804180359206749970813, 3.10745439784079463266932534955, 3.63460253654760430893106204583, 4.39499556117707770816623850890, 5.42089100528302861214854850287, 6.45877964754133375015080142840, 7.55673433754708634935310150831, 8.024148810661537535199895714232, 8.955166620761583546076262497329, 10.04032203908265078567178897451
