| L(s) = 1 | + (0.882 − 1.35i)2-s + (−0.978 + 2.55i)3-s + (−0.254 − 0.571i)4-s + (−1.48 + 1.67i)5-s + (2.60 + 3.58i)6-s + (−2.64 + 0.156i)7-s + (2.19 + 0.348i)8-s + (−3.31 − 2.98i)9-s + (0.963 + 3.49i)10-s + (2.62 + 2.91i)11-s + (1.70 − 0.0894i)12-s + (0.0865 + 0.0441i)13-s + (−2.11 + 3.72i)14-s + (−2.81 − 5.42i)15-s + (3.25 − 3.61i)16-s + (2.10 − 2.60i)17-s + ⋯ |
| L(s) = 1 | + (0.624 − 0.960i)2-s + (−0.565 + 1.47i)3-s + (−0.127 − 0.285i)4-s + (−0.663 + 0.747i)5-s + (1.06 + 1.46i)6-s + (−0.998 + 0.0590i)7-s + (0.777 + 0.123i)8-s + (−1.10 − 0.995i)9-s + (0.304 + 1.10i)10-s + (0.792 + 0.880i)11-s + (0.492 − 0.0258i)12-s + (0.0240 + 0.0122i)13-s + (−0.566 + 0.996i)14-s + (−0.726 − 1.40i)15-s + (0.812 − 0.902i)16-s + (0.511 − 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00694 + 0.603543i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00694 + 0.603543i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.48 - 1.67i)T \) |
| 7 | \( 1 + (2.64 - 0.156i)T \) |
| good | 2 | \( 1 + (-0.882 + 1.35i)T + (-0.813 - 1.82i)T^{2} \) |
| 3 | \( 1 + (0.978 - 2.55i)T + (-2.22 - 2.00i)T^{2} \) |
| 11 | \( 1 + (-2.62 - 2.91i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0865 - 0.0441i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.10 + 2.60i)T + (-3.53 - 16.6i)T^{2} \) |
| 19 | \( 1 + (-5.56 - 2.47i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (4.39 + 2.85i)T + (9.35 + 21.0i)T^{2} \) |
| 29 | \( 1 + (-2.51 + 3.45i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.701 - 0.0737i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.403 - 7.70i)T + (-36.7 + 3.86i)T^{2} \) |
| 41 | \( 1 + (0.124 - 0.0405i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-8.25 - 8.25i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.10 + 1.70i)T + (9.77 - 45.9i)T^{2} \) |
| 53 | \( 1 + (10.9 + 4.21i)T + (39.3 + 35.4i)T^{2} \) |
| 59 | \( 1 + (5.04 + 1.07i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (1.48 + 6.97i)T + (-55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.433 - 0.351i)T + (13.9 + 65.5i)T^{2} \) |
| 71 | \( 1 + (-0.818 - 0.594i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.13 + 0.111i)T + (72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (-15.3 - 1.61i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.69 + 10.6i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (0.981 - 0.208i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (1.77 + 11.1i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.33955517454569723292743328290, −11.87303356453826230189834115128, −11.02620318379328051350927581339, −9.983780985467171176627047258974, −9.667248260126179916580537208199, −7.66573491580925299798753311126, −6.29122868253080160018724904019, −4.74799156999048843927766875750, −3.84967512566606664315808078546, −3.02597264137450965001682698739,
1.06892326778897214817799712151, 3.76337789057090504846470964965, 5.49729210680197975328120820980, 6.16129312682948427460267253429, 7.17065284714163946166037540088, 7.86276854669416323396128456295, 9.204958742267945050358705048212, 10.91795384463692980420477768482, 12.06879438561157024984302441901, 12.59666855447252876271093273597
