L(s) = 1 | + (0.342 − 0.939i)2-s + (−0.402 + 0.0710i)3-s + (−0.766 − 0.642i)4-s + (−0.0710 + 0.402i)6-s + (1.99 + 1.15i)7-s + (−0.866 + 0.500i)8-s + (−2.66 + 0.968i)9-s + (1.32 + 2.29i)11-s + (0.354 + 0.204i)12-s + (−5.02 − 0.885i)13-s + (1.76 − 1.47i)14-s + (0.173 + 0.984i)16-s + (−0.955 + 2.62i)17-s + 2.83i·18-s + (1.11 + 4.21i)19-s + ⋯ |
L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.232 + 0.0409i)3-s + (−0.383 − 0.321i)4-s + (−0.0289 + 0.164i)6-s + (0.753 + 0.434i)7-s + (−0.306 + 0.176i)8-s + (−0.887 + 0.322i)9-s + (0.399 + 0.691i)11-s + (0.102 + 0.0590i)12-s + (−1.39 − 0.245i)13-s + (0.471 − 0.395i)14-s + (0.0434 + 0.246i)16-s + (−0.231 + 0.636i)17-s + 0.667i·18-s + (0.256 + 0.966i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.841751 + 0.547518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.841751 + 0.547518i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.342 + 0.939i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-1.11 - 4.21i)T \) |
good | 3 | \( 1 + (0.402 - 0.0710i)T + (2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-1.99 - 1.15i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.32 - 2.29i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.02 + 0.885i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.955 - 2.62i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.751 + 0.896i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.18 - 1.52i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (4.11 - 7.12i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.99iT - 37T^{2} \) |
| 41 | \( 1 + (-1.97 - 11.2i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.990 - 1.18i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (3.61 + 9.92i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.72 - 3.24i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-10.5 - 3.83i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.53 - 2.12i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.88 - 7.91i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (0.239 - 0.200i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-15.1 + 2.66i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.52 - 8.67i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.53 + 0.887i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.12 - 6.39i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.896 + 2.46i)T + (-74.3 - 62.3i)T^{2} \) |
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
−10.28530533053813028417892477854, −9.531011053844172630857413792600, −8.583435417926079011683578339298, −7.83964054134943417791000992812, −6.72926013659236988310750136819, −5.44911772616044729640214640054, −5.06887223929764477524279668103, −3.91273408024330334848709652994, −2.63206566616469645383531928118, −1.68867854916557704634531254008,
0.43272713942311062953523827518, 2.44420466288414000748521929865, 3.71736287482373144401060194148, 4.83517221229608311445220327155, 5.45902524719225576828852511117, 6.49819833713191836949848264124, 7.33084852394013464279632275355, 8.014984982270887045127330638849, 9.091448497549953153309623669019, 9.552869971174819313287944007152