L(s) = 1 | − 0.618i·3-s + 1.61i·7-s + 2.61·9-s + 11-s − 2.23i·13-s + 1.85i·17-s + 3.47·19-s + 1.00·21-s + 9.32i·23-s − 3.47i·27-s + 6.61·29-s − 0.236·31-s − 0.618i·33-s − 6.23i·37-s − 1.38·39-s + ⋯ |
L(s) = 1 | − 0.356i·3-s + 0.611i·7-s + 0.872·9-s + 0.301·11-s − 0.620i·13-s + 0.449i·17-s + 0.796·19-s + 0.218·21-s + 1.94i·23-s − 0.668i·27-s + 1.22·29-s − 0.0423·31-s − 0.107i·33-s − 1.02i·37-s − 0.221·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.148832629\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.148832629\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 0.618iT - 3T^{2} \) |
| 7 | \( 1 - 1.61iT - 7T^{2} \) |
| 13 | \( 1 + 2.23iT - 13T^{2} \) |
| 17 | \( 1 - 1.85iT - 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 - 9.32iT - 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 0.236T + 31T^{2} \) |
| 37 | \( 1 + 6.23iT - 37T^{2} \) |
| 41 | \( 1 + 9.94T + 41T^{2} \) |
| 43 | \( 1 + 0.472iT - 43T^{2} \) |
| 47 | \( 1 - 8.70iT - 47T^{2} \) |
| 53 | \( 1 - 2.85iT - 53T^{2} \) |
| 59 | \( 1 + 8.23T + 59T^{2} \) |
| 61 | \( 1 + 4.09T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 4.09iT - 73T^{2} \) |
| 79 | \( 1 + 3.85T + 79T^{2} \) |
| 83 | \( 1 - 7.85iT - 83T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 + 16.0iT - 97T^{2} \) |
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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
−8.316072360585865723848411021494, −7.62566795849278481603467388759, −7.08560778473208359010943765877, −6.18486902956298702120597334213, −5.54830341296459386764488868537, −4.76051176864912515962679495865, −3.76727559695501405700724739369, −3.00843873216499746572737861017, −1.87612524074519759405871955715, −1.06094489018478458004194028788,
0.71830089597808218900153775960, 1.78194560279426564051234185310, 2.96426173331069974509170158553, 3.84080024950022784849034837913, 4.61693605083038601328576046099, 5.02729772070537743539690027435, 6.38819152815197519230459530431, 6.79132035901550153788354357498, 7.47557251818254307521404766623, 8.401672161142385390241963704143