| L(s) = 1 | + (0.116 − 1.55i)2-s + (−2.22 + 1.51i)3-s + (−0.425 − 0.0640i)4-s + (−0.322 + 1.41i)5-s + (2.09 + 3.63i)6-s + (−2.55 − 0.681i)7-s + (0.544 − 2.38i)8-s + (1.54 − 3.94i)9-s + (2.15 + 0.665i)10-s + (1.81 − 4.62i)11-s + (1.04 − 0.501i)12-s + (−3.13 − 2.90i)13-s + (−1.35 + 3.89i)14-s + (−1.42 − 3.62i)15-s + (−4.46 − 1.37i)16-s + (−0.263 − 1.15i)17-s + ⋯ |
| L(s) = 1 | + (0.0823 − 1.09i)2-s + (−1.28 + 0.874i)3-s + (−0.212 − 0.0320i)4-s + (−0.144 + 0.631i)5-s + (0.855 + 1.48i)6-s + (−0.966 − 0.257i)7-s + (0.192 − 0.843i)8-s + (0.515 − 1.31i)9-s + (0.682 + 0.210i)10-s + (0.547 − 1.39i)11-s + (0.300 − 0.144i)12-s + (−0.869 − 0.806i)13-s + (−0.362 + 1.04i)14-s + (−0.367 − 0.936i)15-s + (−1.11 − 0.344i)16-s + (−0.0637 − 0.279i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.154898 - 0.520467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.154898 - 0.520467i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.55 + 0.681i)T \) |
| 43 | \( 1 + (1.46 + 6.39i)T \) |
| good | 2 | \( 1 + (-0.116 + 1.55i)T + (-1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (2.22 - 1.51i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.322 - 1.41i)T + (-4.50 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-1.81 + 4.62i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (3.13 + 2.90i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.263 + 1.15i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (-1.00 + 1.26i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (2.26 + 2.83i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (0.710 - 9.48i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (-0.381 + 5.09i)T + (-30.6 - 4.62i)T^{2} \) |
| 37 | \( 1 + 5.88T + 37T^{2} \) |
| 41 | \( 1 + (4.74 - 2.28i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (1.28 + 3.27i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-4.16 - 1.28i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (4.71 - 4.37i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (0.839 + 11.1i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-4.36 - 11.1i)T + (-49.1 + 45.5i)T^{2} \) |
| 71 | \( 1 + (-0.760 - 1.93i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (-1.14 + 5.00i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (6.64 + 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.289 - 3.86i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-13.0 - 6.29i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (0.0497 + 0.0623i)T + (-21.5 + 94.5i)T^{2} \) |
| 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
−11.24173751074149797417193019007, −10.51181289431512697303709910259, −10.12940108976410635577175255468, −9.037429090159380866827022828405, −7.10406004496652816921919876257, −6.31760445797965332815994430851, −5.16113423155276453099228755044, −3.73774348448338916595788911112, −3.01909520567215844931775034179, −0.43697172163564511566970700868,
1.89788767627340782383640402006, 4.51330455521877300164572022666, 5.44559943294101578024651335941, 6.47053157894578856941185660005, 6.91383711511758640505565623693, 7.81014492195827992188749121170, 9.184022760966747853312325107355, 10.19514554476929545976395184238, 11.68752519769193958108485900217, 12.09478047224250840341008669170
