L(s) = 1 | + (−0.491 − 1.32i)2-s + (−1.72 + 0.169i)3-s + (−1.51 + 1.30i)4-s + (0.715 + 0.260i)5-s + (1.07 + 2.20i)6-s + (−3.32 − 0.586i)7-s + (2.47 + 1.37i)8-s + (2.94 − 0.585i)9-s + (−0.00635 − 1.07i)10-s + (1.95 + 5.36i)11-s + (2.39 − 2.50i)12-s + (3.36 + 4.01i)13-s + (0.857 + 4.70i)14-s + (−1.27 − 0.327i)15-s + (0.601 − 3.95i)16-s + (−2.95 + 1.70i)17-s + ⋯ |
L(s) = 1 | + (−0.347 − 0.937i)2-s + (−0.995 + 0.0981i)3-s + (−0.758 + 0.651i)4-s + (0.319 + 0.116i)5-s + (0.437 + 0.899i)6-s + (−1.25 − 0.221i)7-s + (0.874 + 0.484i)8-s + (0.980 − 0.195i)9-s + (−0.00201 − 0.340i)10-s + (0.588 + 1.61i)11-s + (0.690 − 0.723i)12-s + (0.934 + 1.11i)13-s + (0.229 + 1.25i)14-s + (−0.329 − 0.0844i)15-s + (0.150 − 0.988i)16-s + (−0.717 + 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.536990 + 0.167129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.536990 + 0.167129i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.491 + 1.32i)T \) |
| 3 | \( 1 + (1.72 - 0.169i)T \) |
good | 5 | \( 1 + (-0.715 - 0.260i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (3.32 + 0.586i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.95 - 5.36i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.36 - 4.01i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.95 - 1.70i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.27 + 2.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.947 - 5.37i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.27 - 2.74i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.33 + 0.410i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (9.07 - 5.24i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.88 - 2.24i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (5.95 - 2.16i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.625 + 3.54i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 8.06T + 53T^{2} \) |
| 59 | \( 1 + (1.63 - 4.48i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (2.62 + 0.463i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.16 + 1.81i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.03 - 5.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.40 + 2.43i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.42 + 8.84i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.47 + 6.52i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (7.79 + 4.49i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.6 + 4.97i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.13898045441997009053330199733, −11.55422003347763865376339892109, −10.36205603936206503183844562638, −9.776750646010734835334059145240, −8.989038635933080876455923749955, −7.10126035515451377409437575188, −6.41644848992355336584458898033, −4.71566138158207731233776095216, −3.70802450141377277289585091059, −1.70532218013775123961127369404,
0.63701922363362367366180456321, 3.67913879480013159971104100518, 5.42527050639510220407943021239, 6.11090641323347024479334564805, 6.74527410766574048621966774014, 8.261473945765396857024972778698, 9.197776516514582192555218952306, 10.24485273627672003848186048237, 11.02269974273699173464653398398, 12.37299653420943746118337352356