| L(s) = 1 | + (−0.735 + 0.848i)5-s + (0.891 − 0.572i)7-s + (−0.347 + 2.41i)11-s + (1.00 + 0.643i)13-s + (0.810 + 0.237i)17-s + (1.83 − 0.538i)19-s + (0.265 + 4.78i)23-s + (0.531 + 3.70i)25-s + (0.551 + 0.161i)29-s + (0.110 + 0.242i)31-s + (−0.169 + 1.17i)35-s + (5.68 + 6.55i)37-s + (3.19 − 3.68i)41-s + (−1.36 + 2.99i)43-s − 0.561·47-s + ⋯ |
| L(s) = 1 | + (−0.328 + 0.379i)5-s + (0.336 − 0.216i)7-s + (−0.104 + 0.729i)11-s + (0.277 + 0.178i)13-s + (0.196 + 0.0577i)17-s + (0.420 − 0.123i)19-s + (0.0553 + 0.998i)23-s + (0.106 + 0.740i)25-s + (0.102 + 0.0300i)29-s + (0.0198 + 0.0435i)31-s + (−0.0286 + 0.199i)35-s + (0.934 + 1.07i)37-s + (0.498 − 0.575i)41-s + (−0.208 + 0.457i)43-s − 0.0818·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25409 + 0.699991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25409 + 0.699991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-0.265 - 4.78i)T \) |
| good | 5 | \( 1 + (0.735 - 0.848i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.891 + 0.572i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.347 - 2.41i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.00 - 0.643i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.810 - 0.237i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.83 + 0.538i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.551 - 0.161i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.110 - 0.242i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-5.68 - 6.55i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.19 + 3.68i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (1.36 - 2.99i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 0.561T + 47T^{2} \) |
| 53 | \( 1 + (-5.21 + 3.35i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.23 - 0.794i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.69 - 8.08i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.709 - 4.93i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.0399 - 0.277i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (4.93 - 1.45i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (4.92 + 3.16i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (3.38 + 3.90i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.04 + 11.0i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.64 + 3.04i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.31123416005557946343760345013, −9.612614692093233759495717418568, −8.649698173789399518526391050970, −7.60743501075222841305504196244, −7.15100281837295746998442078964, −5.99439953518414157148593955682, −4.95943743541276658248701483480, −3.98272069214493061077299993140, −2.89924028073684723039394684213, −1.43789586693628612978687795193,
0.78718506788446902732807637995, 2.44727195119418490790248233355, 3.67462985567522051576999442510, 4.70665380287734092754774137012, 5.65136089624935900678728820276, 6.54160948561404227778151843473, 7.74213151630574107764999732137, 8.360470145220597671915081279289, 9.097121984608830083442952904342, 10.13706072786282473899835639970
