L(s) = 1 | − 2-s + 4-s − 8-s + (−1.5 + 2.59i)11-s + (−1 + 1.73i)13-s + 16-s + (−1.5 − 2.59i)17-s + (0.5 − 0.866i)19-s + (1.5 − 2.59i)22-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s + (1 − 1.73i)26-s + (3 + 5.19i)29-s − 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.353·8-s + (−0.452 + 0.783i)11-s + (−0.277 + 0.480i)13-s + 0.250·16-s + (−0.363 − 0.630i)17-s + (0.114 − 0.198i)19-s + (0.319 − 0.553i)22-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + (0.196 − 0.339i)26-s + (0.557 + 0.964i)29-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.058556687\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.058556687\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.923651824034210541314266152522, −8.093500395163308513568710186676, −7.23089837584664638546175356192, −6.82712407970766879288589273264, −5.81270967541070715068806227330, −4.84857703102516482141371735729, −4.08403213183626612063990825431, −2.69939529476788285965806656157, −2.08747718039733568228513271301, −0.59159463084271176201386647019,
0.841499227779318127024314506398, 2.09315382817874986257842745361, 3.09595343212417031009774622149, 3.97438039560080224251254609982, 5.24573033346363685696718462661, 5.87058900706700462779698170274, 6.71158262676028955317412597818, 7.65015655580751846149006326545, 8.132841470099691173515657155274, 8.851120212131100275666077497982