| L(s) = 1 | + (−2.03 − 0.841i)2-s + (0.0315 + 0.158i)3-s + (0.590 + 0.590i)4-s + (−4.46 − 2.98i)5-s + (0.0693 − 0.348i)6-s + (5.19 − 3.46i)7-s + (2.66 + 6.42i)8-s + (8.29 − 3.43i)9-s + (6.55 + 9.80i)10-s + (14.3 + 2.84i)11-s + (−0.0749 + 0.112i)12-s + (−4.79 + 4.79i)13-s + (−13.4 + 2.67i)14-s + (0.331 − 0.801i)15-s − 18.6i·16-s + ⋯ |
| L(s) = 1 | + (−1.01 − 0.420i)2-s + (0.0105 + 0.0528i)3-s + (0.147 + 0.147i)4-s + (−0.892 − 0.596i)5-s + (0.0115 − 0.0580i)6-s + (0.741 − 0.495i)7-s + (0.332 + 0.803i)8-s + (0.921 − 0.381i)9-s + (0.655 + 0.980i)10-s + (1.30 + 0.259i)11-s + (−0.00624 + 0.00935i)12-s + (−0.369 + 0.369i)13-s + (−0.961 + 0.191i)14-s + (0.0221 − 0.0534i)15-s − 1.16i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.325065 - 0.676667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.325065 - 0.676667i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (2.03 + 0.841i)T + (2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (-0.0315 - 0.158i)T + (-8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (4.46 + 2.98i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-5.19 + 3.46i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-14.3 - 2.84i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (4.79 - 4.79i)T - 169iT^{2} \) |
| 19 | \( 1 + (23.0 + 9.56i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-2.55 + 12.8i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-18.3 + 27.4i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (-1.19 + 0.236i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (9.06 + 45.5i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (-26.1 + 17.4i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (67.3 - 27.8i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-10.4 + 10.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (4.77 + 1.97i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (10.8 + 26.1i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (45.9 + 68.7i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 44.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-11.3 - 56.8i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (1.12 + 0.748i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-29.1 - 5.79i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-25.7 + 62.2i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (90.1 + 90.1i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (37.5 - 56.2i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−11.18677660648822068883151096969, −10.22411458906104299057507202637, −9.301310362172906068670076611536, −8.550108940452897631432668508315, −7.67359938298174889614117210605, −6.63685370359642567018988210982, −4.62138063285595556979611853320, −4.18319683647514913460576931467, −1.80655643471489009154767394675, −0.58989568339956240346558041282,
1.48489399870757779802385344556, 3.59378024623810580637611781212, 4.65987102744463034823719669943, 6.47609412248706613551440706543, 7.28631607725134216821507905173, 8.116751629479217191385320874092, 8.822633110865180349633060384169, 9.957498451523734369642705612515, 10.81250957357539254892712371868, 11.79442597656770213462596058109
