| L(s) = 1 | − 1.93·2-s − 3.28·3-s + 1.75·4-s − 1.73·5-s + 6.36·6-s + 2.63·7-s + 0.470·8-s + 7.79·9-s + 3.35·10-s + 11-s − 5.77·12-s + 2.10·13-s − 5.11·14-s + 5.69·15-s − 4.42·16-s − 17-s − 15.1·18-s − 7.62·19-s − 3.04·20-s − 8.65·21-s − 1.93·22-s − 0.362·23-s − 1.54·24-s − 1.99·25-s − 4.08·26-s − 15.7·27-s + 4.63·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 1.89·3-s + 0.878·4-s − 0.775·5-s + 2.59·6-s + 0.996·7-s + 0.166·8-s + 2.59·9-s + 1.06·10-s + 0.301·11-s − 1.66·12-s + 0.584·13-s − 1.36·14-s + 1.47·15-s − 1.10·16-s − 0.242·17-s − 3.55·18-s − 1.75·19-s − 0.681·20-s − 1.88·21-s − 0.413·22-s − 0.0755·23-s − 0.315·24-s − 0.399·25-s − 0.801·26-s − 3.02·27-s + 0.875·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2092310820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2092310820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 3 | \( 1 + 3.28T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 - 2.63T + 7T^{2} \) |
| 13 | \( 1 - 2.10T + 13T^{2} \) |
| 19 | \( 1 + 7.62T + 19T^{2} \) |
| 23 | \( 1 + 0.362T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 3.50T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 8.51T + 41T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 14.3T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 7.42T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 7.02T + 97T^{2} \) |
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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
−7.75736581926817621822860640605, −7.34629811917872029046194124924, −6.38915079979659059012385103379, −6.11084912045883195676371206234, −4.93961417946303255734434077091, −4.48806969951548962065488178934, −3.86713519794809193015842917084, −1.97062247203588342167995629398, −1.35797406323867162222323278128, −0.34338355557524854030617402660,
0.34338355557524854030617402660, 1.35797406323867162222323278128, 1.97062247203588342167995629398, 3.86713519794809193015842917084, 4.48806969951548962065488178934, 4.93961417946303255734434077091, 6.11084912045883195676371206234, 6.38915079979659059012385103379, 7.34629811917872029046194124924, 7.75736581926817621822860640605
