L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + 5-s + (2.53 + 0.744i)7-s − 0.999i·8-s + (0.866 − 0.5i)10-s + 0.441i·11-s + (3.17 − 1.83i)13-s + (2.57 − 0.624i)14-s + (−0.5 − 0.866i)16-s + (−0.136 − 0.235i)17-s + (−3.25 − 1.87i)19-s + (0.499 − 0.866i)20-s + (0.220 + 0.382i)22-s − 2.59i·23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447·5-s + (0.959 + 0.281i)7-s − 0.353i·8-s + (0.273 − 0.158i)10-s + 0.133i·11-s + (0.880 − 0.508i)13-s + (0.687 − 0.166i)14-s + (−0.125 − 0.216i)16-s + (−0.0330 − 0.0571i)17-s + (−0.746 − 0.431i)19-s + (0.111 − 0.193i)20-s + (0.0470 + 0.0815i)22-s − 0.541i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.173825895\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.173825895\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.53 - 0.744i)T \) |
good | 11 | \( 1 - 0.441iT - 11T^{2} \) |
| 13 | \( 1 + (-3.17 + 1.83i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.136 + 0.235i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.25 + 1.87i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 2.59iT - 23T^{2} \) |
| 29 | \( 1 + (-2.38 - 1.37i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.57 - 4.37i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0597 + 0.103i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.93 + 6.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.849 + 1.47i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.94 - 6.83i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0822 - 0.0474i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.60 - 2.78i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (11.2 - 6.50i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.268 + 0.465i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.75iT - 71T^{2} \) |
| 73 | \( 1 + (-9.64 + 5.56i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.51 + 2.62i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.29 + 7.43i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.35 + 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.9 + 7.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.958031617513148172546562017161, −8.533227584733792895909284339561, −7.52920193702780225499941132161, −6.49976320938205832762797189098, −5.85335862276844869782115302025, −4.93814532985724430328408331456, −4.33201898778757863283381409235, −3.11202959902267760969627087067, −2.19316955415820445805425045778, −1.12716992069515497186694340263,
1.35111891963227261435592581592, 2.40010730955764469342022234761, 3.68957687523344907318602788329, 4.44745487454777921408377752014, 5.24774846298770429777530062033, 6.19032774104850702865369737386, 6.68626132236211349207226542275, 7.88456676757497494884336908288, 8.272365709183851309230336266822, 9.196551607504913519799071875356