L(s) = 1 | + (−1.12 − 1.94i)2-s + (0.277 + 0.480i)3-s + (−1.52 + 2.64i)4-s + 1.44·5-s + (0.623 − 1.07i)6-s + (1.02 − 1.77i)7-s + 2.35·8-s + (1.34 − 2.33i)9-s + (−1.62 − 2.81i)10-s + (−1.27 − 2.21i)11-s − 1.69·12-s − 4.60·14-s + (0.400 + 0.694i)15-s + (0.400 + 0.694i)16-s + (2.64 − 4.58i)17-s − 6.04·18-s + ⋯ |
L(s) = 1 | + (−0.794 − 1.37i)2-s + (0.160 + 0.277i)3-s + (−0.762 + 1.32i)4-s + 0.646·5-s + (0.254 − 0.440i)6-s + (0.387 − 0.670i)7-s + 0.833·8-s + (0.448 − 0.777i)9-s + (−0.513 − 0.889i)10-s + (−0.385 − 0.667i)11-s − 0.488·12-s − 1.23·14-s + (0.103 + 0.179i)15-s + (0.100 + 0.173i)16-s + (0.642 − 1.11i)17-s − 1.42·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.468862 - 0.746537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.468862 - 0.746537i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (1.12 + 1.94i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.277 - 0.480i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 + (-1.02 + 1.77i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.27 + 2.21i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.64 + 4.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.92 - 5.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.945 - 1.63i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.13 + 1.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.26T + 31T^{2} \) |
| 37 | \( 1 + (-2.67 - 4.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.637 - 1.10i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.06 - 5.31i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 + (6.10 - 10.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.28 - 7.41i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.288 - 0.499i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.29 - 3.97i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 7.72T + 83T^{2} \) |
| 89 | \( 1 + (-3.30 - 5.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.96 + 10.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.15836106979494755575747082313, −11.33204111691160329989223093198, −10.19863433196619759390761616131, −9.854957865324813564497159491176, −8.773825595905640774039793309036, −7.66314262783182347160004044771, −5.98004456164209953798977206940, −4.17350699102445290711291341779, −2.92403365456373249923848739359, −1.21457931513965198247650443666,
2.10733452583127623776427609752, 4.87754023786801768500699109148, 5.88007385283961264644554586828, 6.98069586484077288246712179424, 7.945879412669335960905066978024, 8.734142681309732121573547923881, 9.789333694698740747997270509197, 10.70270267797774437592931441116, 12.34513401850382240147921692645, 13.30274802493991943642062769430