L(s) = 1 | + 2.65·2-s + 3-s + 5.04·4-s + 2.16·5-s + 2.65·6-s − 2.02·7-s + 8.08·8-s + 9-s + 5.74·10-s + 11-s + 5.04·12-s + 0.318·13-s − 5.36·14-s + 2.16·15-s + 11.3·16-s − 1.66·17-s + 2.65·18-s + 0.0856·19-s + 10.9·20-s − 2.02·21-s + 2.65·22-s − 6.29·23-s + 8.08·24-s − 0.318·25-s + 0.844·26-s + 27-s − 10.2·28-s + ⋯ |
L(s) = 1 | + 1.87·2-s + 0.577·3-s + 2.52·4-s + 0.967·5-s + 1.08·6-s − 0.764·7-s + 2.85·8-s + 0.333·9-s + 1.81·10-s + 0.301·11-s + 1.45·12-s + 0.0882·13-s − 1.43·14-s + 0.558·15-s + 2.84·16-s − 0.404·17-s + 0.625·18-s + 0.0196·19-s + 2.44·20-s − 0.441·21-s + 0.565·22-s − 1.31·23-s + 1.65·24-s − 0.0636·25-s + 0.165·26-s + 0.192·27-s − 1.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.578200521\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.578200521\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 - 2.65T + 2T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 13 | \( 1 - 0.318T + 13T^{2} \) |
| 17 | \( 1 + 1.66T + 17T^{2} \) |
| 19 | \( 1 - 0.0856T + 19T^{2} \) |
| 23 | \( 1 + 6.29T + 23T^{2} \) |
| 29 | \( 1 + 2.14T + 29T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 + 8.58T + 37T^{2} \) |
| 41 | \( 1 - 3.06T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 + 4.56T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 4.06T + 71T^{2} \) |
| 73 | \( 1 - 5.81T + 73T^{2} \) |
| 79 | \( 1 - 7.59T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−9.347230264143018911435673970915, −8.204277549196563907367024339611, −7.13956647564677643299612505539, −6.53470985942560369017080303474, −5.84930865440433706005291581692, −5.18737252535529314062657793811, −4.03093116736718723536018626472, −3.53171042654234310072302671510, −2.44803338706672704579482096654, −1.82392696907191059190234085463,
1.82392696907191059190234085463, 2.44803338706672704579482096654, 3.53171042654234310072302671510, 4.03093116736718723536018626472, 5.18737252535529314062657793811, 5.84930865440433706005291581692, 6.53470985942560369017080303474, 7.13956647564677643299612505539, 8.204277549196563907367024339611, 9.347230264143018911435673970915