| L(s) = 1 | + (−1.93 − 0.203i)2-s + (−0.500 + 2.35i)3-s + (1.75 + 0.373i)4-s + (−0.306 + 2.21i)5-s + (1.45 − 4.46i)6-s + (−2.52 + 0.802i)7-s + (0.377 + 0.122i)8-s + (−2.56 − 1.14i)9-s + (1.04 − 4.22i)10-s + (2.26 − 1.00i)11-s + (−1.76 + 3.95i)12-s + (−2.29 − 3.16i)13-s + (5.04 − 1.04i)14-s + (−5.06 − 1.83i)15-s + (−3.98 − 1.77i)16-s + (−2.30 + 2.07i)17-s + ⋯ |
| L(s) = 1 | + (−1.37 − 0.144i)2-s + (−0.289 + 1.36i)3-s + (0.878 + 0.186i)4-s + (−0.137 + 0.990i)5-s + (0.592 − 1.82i)6-s + (−0.952 + 0.303i)7-s + (0.133 + 0.0433i)8-s + (−0.854 − 0.380i)9-s + (0.330 − 1.33i)10-s + (0.683 − 0.304i)11-s + (−0.508 + 1.14i)12-s + (−0.637 − 0.876i)13-s + (1.34 − 0.278i)14-s + (−1.30 − 0.472i)15-s + (−0.997 − 0.443i)16-s + (−0.558 + 0.502i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0220152 - 0.277325i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0220152 - 0.277325i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.306 - 2.21i)T \) |
| 7 | \( 1 + (2.52 - 0.802i)T \) |
| good | 2 | \( 1 + (1.93 + 0.203i)T + (1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (0.500 - 2.35i)T + (-2.74 - 1.22i)T^{2} \) |
| 11 | \( 1 + (-2.26 + 1.00i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (2.29 + 3.16i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.30 - 2.07i)T + (1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (5.00 - 1.06i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-4.53 - 0.476i)T + (22.4 + 4.78i)T^{2} \) |
| 29 | \( 1 + (0.797 + 2.45i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.22 - 4.68i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-3.31 + 7.45i)T + (-24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (4.21 - 3.06i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.80iT - 43T^{2} \) |
| 47 | \( 1 + (1.61 + 1.45i)T + (4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (1.59 - 7.51i)T + (-48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.15 - 10.9i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (0.904 - 8.60i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (6.37 - 5.73i)T + (7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-4.36 - 13.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (5.00 + 11.2i)T + (-48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (4.18 - 4.64i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (4.78 + 1.55i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.45 + 13.8i)T + (-87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (-4.30 + 1.39i)T + (78.4 - 57.0i)T^{2} \) |
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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
−13.12032663179534705741128062024, −11.66440954314056532569330769147, −10.66251076640813573904393715394, −10.28778477419286522511078935707, −9.434152473644546413389739906349, −8.584596368519048995415364905433, −7.16393189746003631845950389847, −6.00563367557171652209663636521, −4.27315562640448709378033269198, −2.86434998435255576309442903969,
0.40408292363842818713151899360, 1.86447933462948192018786262382, 4.51365107214742066804641230000, 6.57067602543534170820778710031, 6.95862914668735823054224758928, 8.130507436446994962723549901074, 9.099976160287067659101188505294, 9.778669961186532460850247794936, 11.26608343130505456841609132538, 12.21614996041467291788905158993
