L(s) = 1 | + (−1.74 − 3.01i)5-s + (−1.34 + 2.33i)7-s + (−2.84 + 4.92i)11-s + (−1.76 − 3.04i)13-s − 7.65·17-s + 2.02·19-s + (0.0370 + 0.0642i)23-s + (−3.57 + 6.18i)25-s + (2.46 − 4.26i)29-s + (3.72 + 6.44i)31-s + 9.40·35-s + 5.00·37-s + (−0.482 − 0.834i)41-s + (0.255 − 0.442i)43-s + (2.83 − 4.91i)47-s + ⋯ |
L(s) = 1 | + (−0.779 − 1.34i)5-s + (−0.510 + 0.883i)7-s + (−0.857 + 1.48i)11-s + (−0.488 − 0.845i)13-s − 1.85·17-s + 0.463·19-s + (0.00773 + 0.0133i)23-s + (−0.714 + 1.23i)25-s + (0.456 − 0.791i)29-s + (0.668 + 1.15i)31-s + 1.59·35-s + 0.823·37-s + (−0.0752 − 0.130i)41-s + (0.0389 − 0.0674i)43-s + (0.413 − 0.716i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9006880902\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9006880902\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.74 + 3.01i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.34 - 2.33i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.84 - 4.92i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.76 + 3.04i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 2.02T + 19T^{2} \) |
| 23 | \( 1 + (-0.0370 - 0.0642i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.46 + 4.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.72 - 6.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.00T + 37T^{2} \) |
| 41 | \( 1 + (0.482 + 0.834i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.255 + 0.442i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.83 + 4.91i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + (4.47 + 7.75i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.46 + 2.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.56 + 2.71i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 - 5.21T + 73T^{2} \) |
| 79 | \( 1 + (0.716 - 1.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.74 + 3.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + (3.50 - 6.06i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.586854498635634041291782335767, −7.88837262612039616108098275542, −7.21878769838415093051130678548, −6.28740178638338360598933172528, −5.21024401894322169145758416766, −4.83252567251934406289050067302, −4.11697171312819552989467878441, −2.81146811164743128344802797548, −2.06646964538614898770278714334, −0.50976457807393461283039191112,
0.55635342173966332077995012338, 2.46027542557850100431038961366, 3.02972142668651868034613238783, 3.93324989609313494862208582010, 4.54517495607440742385456608179, 5.86788412837524744250141183422, 6.57445689878267277017262725932, 7.12415927343660330958100022070, 7.71684266060638658149191875716, 8.558097555674328765064131759861