L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 4.59·7-s − 0.999·8-s + (−0.499 + 0.866i)10-s − 0.473·11-s + (0.263 − 0.455i)13-s + (−2.29 − 3.98i)14-s + (−0.5 − 0.866i)16-s + (−2.27 − 3.93i)17-s + (2.56 + 3.52i)19-s − 0.999·20-s + (−0.236 − 0.410i)22-s + (−0.236 + 0.410i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 1.73·7-s − 0.353·8-s + (−0.158 + 0.273i)10-s − 0.142·11-s + (0.0730 − 0.126i)13-s + (−0.614 − 1.06i)14-s + (−0.125 − 0.216i)16-s + (−0.551 − 0.954i)17-s + (0.587 + 0.808i)19-s − 0.223·20-s + (−0.0504 − 0.0874i)22-s + (−0.0493 + 0.0855i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6764415603\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6764415603\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-2.56 - 3.52i)T \) |
good | 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 + 0.473T + 11T^{2} \) |
| 13 | \( 1 + (-0.263 + 0.455i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.27 + 3.93i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (0.236 - 0.410i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.32 + 7.49i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.526T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (5.63 + 9.75i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.33 + 10.9i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.07 + 7.05i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.964 + 1.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.53 - 2.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.53 - 7.85i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.27 + 3.93i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.46 - 2.53i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.736 - 1.27i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.10T + 83T^{2} \) |
| 89 | \( 1 + (0.964 - 1.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.79 + 15.2i)T + (-48.5 + 84.0i)T^{2} \) |
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
−9.162534690391240051861603323339, −8.408157591399298354417294850457, −7.17334484341148813758889137514, −6.90717342451600671145415579139, −5.95728008336597280089541399545, −5.38793715392485758088766396122, −4.05629774560629155614909858357, −3.30761891089830727333474717912, −2.42193136415198379481636951675, −0.23015875231207545739589979128,
1.31216249679929664331950819900, 2.75022171422650488970917553252, 3.36303570407595423943006407148, 4.43050879506436436806433867890, 5.30213969784561633195628648413, 6.40201857460489790739428196621, 6.66484577030016766403761520352, 8.042090530661626768822194697251, 9.034622252276630585078550564415, 9.483384838558699048505110320309