| L(s) = 1 | + (−1.90 + 0.335i)2-s + (1.88 + 2.33i)3-s + (−0.255 + 0.0929i)4-s + (5.49 + 6.54i)5-s + (−4.36 − 3.81i)6-s + (1.56 + 2.71i)7-s + (7.14 − 4.12i)8-s + (−1.92 + 8.79i)9-s + (−12.6 − 10.6i)10-s − 8.67i·11-s + (−0.697 − 0.422i)12-s + (5.85 + 4.91i)13-s + (−3.89 − 4.63i)14-s + (−4.97 + 25.1i)15-s + (−11.3 + 9.53i)16-s + (−13.6 − 16.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.950 + 0.167i)2-s + (0.626 + 0.779i)3-s + (−0.0638 + 0.0232i)4-s + (1.09 + 1.30i)5-s + (−0.726 − 0.635i)6-s + (0.223 + 0.387i)7-s + (0.892 − 0.515i)8-s + (−0.213 + 0.976i)9-s + (−1.26 − 1.06i)10-s − 0.788i·11-s + (−0.0581 − 0.0351i)12-s + (0.450 + 0.377i)13-s + (−0.277 − 0.331i)14-s + (−0.331 + 1.67i)15-s + (−0.710 + 0.596i)16-s + (−0.803 − 0.957i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.600347 + 1.04583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.600347 + 1.04583i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.88 - 2.33i)T \) |
| 19 | \( 1 + (-13.0 - 13.7i)T \) |
| good | 2 | \( 1 + (1.90 - 0.335i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-5.49 - 6.54i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-1.56 - 2.71i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 8.67iT - 121T^{2} \) |
| 13 | \( 1 + (-5.85 - 4.91i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (13.6 + 16.2i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (2.66 + 7.32i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (0.136 + 0.376i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + 36.3T + 961T^{2} \) |
| 37 | \( 1 - 8.21T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-64.4 + 11.3i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (24.9 + 9.09i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (4.49 + 12.3i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (2.84 + 0.502i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-36.9 + 101. i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (5.12 + 4.29i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (1.26 - 7.14i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-42.1 + 7.43i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-83.2 - 30.2i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (81.0 - 68.0i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-111. + 64.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-47.6 - 130. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-8.53 - 48.3i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−13.30554646296105472072108810723, −11.24359421356379606716731395329, −10.60319991752896784899994310516, −9.611035389338293080923191008555, −9.087502849207741981501612473792, −7.975967380089385015998029188717, −6.76436006588719454867899808889, −5.39140950068831867533861421454, −3.64455432090494636787624895334, −2.22787777290510041009860633257,
1.03467077592746780675326935090, 2.00085453835501073249318052688, 4.45642390431321164553551933502, 5.79650339277103138101230081283, 7.34884320028276298333209671888, 8.368706568524802111609134800650, 9.129447336251734219820907502752, 9.726429483903345619067755153011, 10.96601662538454896926490941941, 12.49557889072437148166188843790
