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 | \( 1 + (-0.896 - 2.46i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−9.552869971174819313287944007152, −9.091448497549953153309623669019, −8.014984982270887045127330638849, −7.33084852394013464279632275355, −6.49819833713191836949848264124, −5.45902524719225576828852511117, −4.83517221229608311445220327155, −3.71736287482373144401060194148, −2.44420466288414000748521929865, −0.43272713942311062953523827518,
1.68867854916557704634531254008, 2.63206566616469645383531928118, 3.91273408024330334848709652994, 5.06887223929764477524279668103, 5.44911772616044729640214640054, 6.72926013659236988310750136819, 7.83964054134943417791000992812, 8.583435417926079011683578339298, 9.531011053844172630857413792600, 10.28530533053813028417892477854