| L(s) = 1 | + (−1.39 + 2.26i)2-s + (0.723 + 1.05i)3-s + (−2.26 − 4.53i)4-s + (1.26 − 1.51i)5-s + (−3.40 + 0.157i)6-s + (1.95 − 0.864i)7-s + (8.13 + 0.753i)8-s + (0.491 − 1.26i)9-s + (1.66 + 4.97i)10-s + (0.977 + 0.327i)11-s + (3.16 − 5.67i)12-s + (−0.477 + 5.15i)13-s + (−0.786 + 5.63i)14-s + (2.51 + 0.233i)15-s + (−6.97 + 9.23i)16-s + (4.05 − 0.747i)17-s + ⋯ |
| L(s) = 1 | + (−0.989 + 1.59i)2-s + (0.417 + 0.610i)3-s + (−1.13 − 2.26i)4-s + (0.563 − 0.678i)5-s + (−1.38 + 0.0642i)6-s + (0.740 − 0.326i)7-s + (2.87 + 0.266i)8-s + (0.163 − 0.422i)9-s + (0.527 + 1.57i)10-s + (0.294 + 0.0987i)11-s + (0.912 − 1.63i)12-s + (−0.132 + 1.42i)13-s + (−0.210 + 1.50i)14-s + (0.649 + 0.0601i)15-s + (−1.74 + 2.30i)16-s + (0.983 − 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0710 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0710 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.699305 + 0.750863i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.699305 + 0.750863i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-4.05 + 0.747i)T \) |
| good | 2 | \( 1 + (1.39 - 2.26i)T + (-0.891 - 1.79i)T^{2} \) |
| 3 | \( 1 + (-0.723 - 1.05i)T + (-1.08 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-1.26 + 1.51i)T + (-0.918 - 4.91i)T^{2} \) |
| 7 | \( 1 + (-1.95 + 0.864i)T + (4.71 - 5.17i)T^{2} \) |
| 11 | \( 1 + (-0.977 - 0.327i)T + (8.77 + 6.62i)T^{2} \) |
| 13 | \( 1 + (0.477 - 5.15i)T + (-12.7 - 2.38i)T^{2} \) |
| 19 | \( 1 + (2.11 + 3.41i)T + (-8.46 + 17.0i)T^{2} \) |
| 23 | \( 1 + (-0.984 + 0.434i)T + (15.4 - 16.9i)T^{2} \) |
| 29 | \( 1 + (-2.62 + 7.82i)T + (-23.1 - 17.4i)T^{2} \) |
| 31 | \( 1 + (3.74 - 3.11i)T + (5.69 - 30.4i)T^{2} \) |
| 37 | \( 1 + (5.97 - 3.32i)T + (19.4 - 31.4i)T^{2} \) |
| 41 | \( 1 + (4.66 - 3.19i)T + (14.8 - 38.2i)T^{2} \) |
| 43 | \( 1 + (0.303 - 0.229i)T + (11.7 - 41.3i)T^{2} \) |
| 47 | \( 1 + (-6.47 + 2.50i)T + (34.7 - 31.6i)T^{2} \) |
| 53 | \( 1 + (3.76 - 9.72i)T + (-39.1 - 35.7i)T^{2} \) |
| 59 | \( 1 + (-2.77 - 3.04i)T + (-5.44 + 58.7i)T^{2} \) |
| 61 | \( 1 + (-5.25 + 0.242i)T + (60.7 - 5.62i)T^{2} \) |
| 67 | \( 1 + (-7.66 + 4.74i)T + (29.8 - 59.9i)T^{2} \) |
| 71 | \( 1 + (5.12 + 11.6i)T + (-47.8 + 52.4i)T^{2} \) |
| 73 | \( 1 + (9.92 - 1.38i)T + (70.2 - 19.9i)T^{2} \) |
| 79 | \( 1 + (0.986 - 4.19i)T + (-70.7 - 35.2i)T^{2} \) |
| 83 | \( 1 + (2.98 + 15.9i)T + (-77.3 + 29.9i)T^{2} \) |
| 89 | \( 1 + (-1.46 - 15.7i)T + (-87.4 + 16.3i)T^{2} \) |
| 97 | \( 1 + (0.277 + 0.627i)T + (-65.3 + 71.6i)T^{2} \) |
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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
−11.94268474058069774021123978579, −10.54980475668896932524437592272, −9.592891937601985968345766159101, −9.128712487253651063506065276029, −8.393876056505853721068931042191, −7.24728006645378010871974823925, −6.37635186596411796114048033673, −5.10734613140124409689643117969, −4.33029716576025884513084407941, −1.37244627163441865453889159582,
1.47107374135401093729824288062, 2.45737434870085747122164010082, 3.51044901195255994668306239039, 5.33487259460917207067484054876, 7.19680520067684131361649636283, 8.124844308495012334625522187929, 8.675889179269896552995505836658, 10.12074366171725714292816925132, 10.41745327325253202010402894155, 11.35993685141715653036570612621
