| L(s) = 1 | + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−1.29 + 0.832i)5-s + (−0.654 + 0.755i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.219 − 1.52i)10-s + (1.17 + 2.58i)11-s + (−0.415 − 0.909i)12-s + (−0.0245 − 0.170i)13-s + (−0.841 − 0.540i)14-s + (−1.47 + 0.433i)15-s + (−0.142 + 0.989i)16-s + (−1.74 + 2.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.579 + 0.372i)5-s + (−0.267 + 0.308i)6-s + (−0.0537 + 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0693 − 0.482i)10-s + (0.355 + 0.778i)11-s + (−0.119 − 0.262i)12-s + (−0.00679 − 0.0472i)13-s + (−0.224 − 0.144i)14-s + (−0.381 + 0.112i)15-s + (−0.0355 + 0.247i)16-s + (−0.423 + 0.488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.168689 + 1.01571i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.168689 + 1.01571i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (-3.84 - 2.86i)T \) |
| good | 5 | \( 1 + (1.29 - 0.832i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 2.58i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.0245 + 0.170i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (1.74 - 2.01i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.25 + 2.60i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.291 + 0.336i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (6.36 - 1.86i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (1.80 + 1.16i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (4.38 - 2.82i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (0.783 + 0.229i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 3.64T + 47T^{2} \) |
| 53 | \( 1 + (1.90 - 13.2i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.191 - 1.33i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-1.37 + 0.404i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (4.67 - 10.2i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.44 + 3.17i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (6.75 + 7.79i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.26 + 8.80i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-8.11 - 5.21i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-13.4 - 3.93i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-9.45 + 6.07i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.26976101470398989882059398935, −9.195229666272110805962515851372, −8.880840353878873250841087638269, −7.75932249881640407882580731830, −7.19443181345617960788336645974, −6.35616721354750615812807572491, −5.15682483612547190053197302257, −4.22272388097682455563766014234, −3.19473866850565408822985696199, −1.79451396235884686976244632093,
0.49020943988707804835687724704, 1.93069119418345940537044600425, 3.24882829959858691386848684112, 3.98052982290027306131061031888, 4.97595552654914635645407484939, 6.40250485734142954066036246193, 7.32428286103707432951812582073, 8.279760449660659553800652238321, 8.719814861710064229135658308439, 9.587011937477267365789324207204
