| L(s) = 1 | + (0.830 − 1.81i)3-s + (−0.654 + 0.755i)5-s + (2.04 − 1.31i)7-s + (−0.650 − 0.750i)9-s + (0.564 − 3.92i)11-s + (−1.16 − 0.747i)13-s + (0.830 + 1.81i)15-s + (−1.96 − 0.578i)17-s + (5.29 − 1.55i)19-s + (−0.691 − 4.81i)21-s + (−2.85 + 3.85i)23-s + (−0.142 − 0.989i)25-s + (3.84 − 1.12i)27-s + (1.05 + 0.309i)29-s + (−2.31 − 5.06i)31-s + ⋯ |
| L(s) = 1 | + (0.479 − 1.04i)3-s + (−0.292 + 0.337i)5-s + (0.773 − 0.496i)7-s + (−0.216 − 0.250i)9-s + (0.170 − 1.18i)11-s + (−0.322 − 0.207i)13-s + (0.214 + 0.469i)15-s + (−0.477 − 0.140i)17-s + (1.21 − 0.356i)19-s + (−0.150 − 1.04i)21-s + (−0.595 + 0.803i)23-s + (−0.0284 − 0.197i)25-s + (0.740 − 0.217i)27-s + (0.195 + 0.0575i)29-s + (−0.415 − 0.910i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27745 - 1.06414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27745 - 1.06414i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (2.85 - 3.85i)T \) |
| good | 3 | \( 1 + (-0.830 + 1.81i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (-2.04 + 1.31i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.564 + 3.92i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.16 + 0.747i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.96 + 0.578i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-5.29 + 1.55i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.05 - 0.309i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.31 + 5.06i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.0422 + 0.0487i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.21 + 2.56i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.959 - 2.10i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 7.68T + 47T^{2} \) |
| 53 | \( 1 + (4.52 - 2.90i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-7.86 - 5.05i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.66 - 5.82i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.18 - 8.22i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.498 - 3.46i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-8.60 + 2.52i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-11.4 - 7.36i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (0.379 + 0.438i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (6.38 - 13.9i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.97 + 3.43i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.18531654674644613206470335058, −9.977827394611924125931041242321, −8.802635918146110513211852784254, −7.85001524702308797871481585444, −7.46571851839391934252497605248, −6.40873147535341430127438587577, −5.18000628380966761386690594997, −3.77404364129088941804683350361, −2.54415069407805808493362303323, −1.10018183131486521256234122215,
1.94030536522115863421719144360, 3.47495235042782223468338853563, 4.59025301215650017032786401550, 5.08922609880564482842027709015, 6.69101512090753170889152774722, 7.84630198833617524208612220722, 8.671801815813300769822687473537, 9.531211004138765781349481292619, 10.11331407553588628987759807074, 11.23083228391803805057871159296
