| L(s) = 1 | + (0.610 + 1.27i)2-s + (−0.773 − 0.634i)3-s + (−1.25 + 1.55i)4-s + (−0.567 − 1.87i)5-s + (0.337 − 1.37i)6-s + (−2.34 − 0.466i)7-s + (−2.75 − 0.651i)8-s + (0.195 + 0.980i)9-s + (2.04 − 1.86i)10-s + (−0.302 + 3.06i)11-s + (1.95 − 0.407i)12-s + (−3.97 − 1.20i)13-s + (−0.836 − 3.27i)14-s + (−0.748 + 1.80i)15-s + (−0.847 − 3.90i)16-s + (−2.71 − 6.54i)17-s + ⋯ |
| L(s) = 1 | + (0.431 + 0.902i)2-s + (−0.446 − 0.366i)3-s + (−0.627 + 0.778i)4-s + (−0.253 − 0.837i)5-s + (0.137 − 0.560i)6-s + (−0.886 − 0.176i)7-s + (−0.973 − 0.230i)8-s + (0.0650 + 0.326i)9-s + (0.645 − 0.590i)10-s + (−0.0910 + 0.924i)11-s + (0.565 − 0.117i)12-s + (−1.10 − 0.334i)13-s + (−0.223 − 0.876i)14-s + (−0.193 + 0.466i)15-s + (−0.211 − 0.977i)16-s + (−0.657 − 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.133889 - 0.215913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133889 - 0.215913i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.610 - 1.27i)T \) |
| 3 | \( 1 + (0.773 + 0.634i)T \) |
| good | 5 | \( 1 + (0.567 + 1.87i)T + (-4.15 + 2.77i)T^{2} \) |
| 7 | \( 1 + (2.34 + 0.466i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (0.302 - 3.06i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (3.97 + 1.20i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (2.71 + 6.54i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (2.55 + 4.78i)T + (-10.5 + 15.7i)T^{2} \) |
| 23 | \( 1 + (3.04 - 4.55i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (4.47 - 0.440i)T + (28.4 - 5.65i)T^{2} \) |
| 31 | \( 1 + (-6.83 - 6.83i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.99 - 1.06i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (-8.07 - 5.39i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-4.18 + 3.43i)T + (8.38 - 42.1i)T^{2} \) |
| 47 | \( 1 + (8.63 - 3.57i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (13.6 + 1.34i)T + (51.9 + 10.3i)T^{2} \) |
| 59 | \( 1 + (8.25 - 2.50i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-5.56 + 6.78i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (0.0285 - 0.0347i)T + (-13.0 - 65.7i)T^{2} \) |
| 71 | \( 1 + (-0.188 + 0.946i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.90 + 0.378i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (1.72 + 0.716i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.63 + 0.873i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-4.29 - 6.42i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (3.35 + 3.35i)T + 97iT^{2} \) |
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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
−11.36659798886234332842953012164, −9.741851018392655989044487874769, −9.238969377392675674164349008714, −7.918146580460957305037869675123, −7.14055649166343409124820428420, −6.40547579961847677216131835163, −4.97844884956545666367395356752, −4.60922815854863493096635116498, −2.83145817316555271925809089565, −0.14548940261024061215303749784,
2.37037307980829512649254438308, 3.54909024395474589966795528460, 4.41340523593468555003286109307, 6.02541958343952090533126721548, 6.34132462605188796786367783025, 8.061166101661555848644160665718, 9.280785537098679746893971161750, 10.14088090832692748926842764422, 10.76161934659511591563783301671, 11.49269649716146664782286224519
