| L(s) = 1 | + 141.·7-s − 273.·11-s − 945.·13-s + 1.21e3·17-s + 135.·19-s − 2.73e3·23-s + 3.07e3·29-s − 1.41e3·31-s + 3.15e3·37-s + 1.34e4·41-s − 5.43e3·43-s − 3.26e3·47-s + 3.17e3·49-s + 2.14e4·53-s + 4.14e4·59-s − 2.43e4·61-s + 3.23e4·67-s + 3.86e4·71-s − 8.08e3·73-s − 3.86e4·77-s − 1.59e4·79-s + 4.84e4·83-s − 8.91e4·89-s − 1.33e5·91-s + 5.68e4·97-s − 6.74e4·101-s − 6.17e4·103-s + ⋯ |
| L(s) = 1 | + 1.09·7-s − 0.681·11-s − 1.55·13-s + 1.01·17-s + 0.0863·19-s − 1.07·23-s + 0.679·29-s − 0.263·31-s + 0.378·37-s + 1.25·41-s − 0.448·43-s − 0.215·47-s + 0.188·49-s + 1.04·53-s + 1.54·59-s − 0.838·61-s + 0.879·67-s + 0.910·71-s − 0.177·73-s − 0.742·77-s − 0.288·79-s + 0.772·83-s − 1.19·89-s − 1.69·91-s + 0.613·97-s − 0.657·101-s − 0.573·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.121440601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.121440601\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 141.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 273.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 945.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.21e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 135.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.43e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.14e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.43e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.86e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.08e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.68e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.538435127469695109381263515418, −8.245747277968842431199308274346, −7.82767814972882461347332286896, −6.99974623610932330752541486320, −5.66808747575479107698227638958, −5.04215206409438834657304569587, −4.15234381479750751772489625152, −2.78564206852036039591676183395, −1.92059336780640908404134633625, −0.64162286140495330432022461620,
0.64162286140495330432022461620, 1.92059336780640908404134633625, 2.78564206852036039591676183395, 4.15234381479750751772489625152, 5.04215206409438834657304569587, 5.66808747575479107698227638958, 6.99974623610932330752541486320, 7.82767814972882461347332286896, 8.245747277968842431199308274346, 9.538435127469695109381263515418
