L(s) = 1 | + (1.38 − 2.39i)5-s + 3.26i·7-s + 1.94i·11-s + (3.96 − 2.29i)13-s + (−2.27 + 3.94i)17-s + (3.43 + 2.68i)19-s + (−7.81 + 4.51i)23-s + (−1.32 − 2.29i)25-s + (1.78 − 1.03i)29-s + 1.21·31-s + (7.81 + 4.51i)35-s + 1.73i·37-s + (3.25 + 1.87i)41-s + (1.82 + 1.05i)43-s + (−9.49 + 5.48i)47-s + ⋯ |
L(s) = 1 | + (0.618 − 1.07i)5-s + 1.23i·7-s + 0.585i·11-s + (1.10 − 0.635i)13-s + (−0.551 + 0.955i)17-s + (0.787 + 0.615i)19-s + (−1.62 + 0.940i)23-s + (−0.264 − 0.458i)25-s + (0.331 − 0.191i)29-s + 0.218·31-s + (1.32 + 0.762i)35-s + 0.284i·37-s + (0.508 + 0.293i)41-s + (0.278 + 0.160i)43-s + (−1.38 + 0.799i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.905700399\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.905700399\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.43 - 2.68i)T \) |
good | 5 | \( 1 + (-1.38 + 2.39i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 3.26iT - 7T^{2} \) |
| 11 | \( 1 - 1.94iT - 11T^{2} \) |
| 13 | \( 1 + (-3.96 + 2.29i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.27 - 3.94i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (7.81 - 4.51i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.78 + 1.03i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 - 1.73iT - 37T^{2} \) |
| 41 | \( 1 + (-3.25 - 1.87i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.82 - 1.05i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.49 - 5.48i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.40 - 4.27i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.68 + 2.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.322 - 0.559i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.04 - 7.00i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.13 + 10.6i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.79 - 6.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.04 - 8.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.9iT - 83T^{2} \) |
| 89 | \( 1 + (-5.93 + 3.42i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (16.4 + 9.47i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.891892298911910943908262168190, −8.300533995568110907467773993376, −7.73232113819640626177039828875, −6.22508073689840769407201207410, −5.92615089894121792154487305058, −5.20861054902457005724152124435, −4.30243785904005565344527250888, −3.27281386012503751392154412136, −2.03106004010156840131314052695, −1.34187179903746339523274342514,
0.63442826372592769321116166050, 1.99433989152747835784265039742, 3.01926943153228543807255940403, 3.83041199641482682773045355559, 4.66782069922343567665358586364, 5.83385245700262853728670110044, 6.60954368315089834996433797854, 6.92578217360281679910730805930, 7.88401990599902061278543487360, 8.694066017845791018321603426431