| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 6·11-s + 12-s − 2·13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 2·19-s + 20-s − 21-s − 6·22-s + 4·23-s − 24-s + 25-s + 2·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.218·21-s − 1.27·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.444192882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.444192882\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.82806960419972, −13.02162777386651, −12.74364839381720, −12.22626196823301, −11.52023883301771, −11.26591584903252, −10.69441918565526, −9.937761830176429, −9.567107968328143, −9.259264255911144, −8.956124573025672, −8.306147732593149, −7.593783146988493, −7.300943933922205, −6.635785386106976, −6.298308482594574, −5.714611742748897, −4.975799333324204, −4.201940030494370, −3.834356182424799, −3.050580042954719, −2.532295385736511, −1.928157770091376, −1.148915831093911, −0.7028588819627286,
0.7028588819627286, 1.148915831093911, 1.928157770091376, 2.532295385736511, 3.050580042954719, 3.834356182424799, 4.201940030494370, 4.975799333324204, 5.714611742748897, 6.298308482594574, 6.635785386106976, 7.300943933922205, 7.593783146988493, 8.306147732593149, 8.956124573025672, 9.259264255911144, 9.567107968328143, 9.937761830176429, 10.69441918565526, 11.26591584903252, 11.52023883301771, 12.22626196823301, 12.74364839381720, 13.02162777386651, 13.82806960419972
