| L(s) = 1 | + (−4.65 + 2.68i)5-s + (−4.29 + 7.44i)11-s − 21.0i·13-s + (−4.75 − 2.74i)17-s + (−6.27 + 3.62i)19-s + (14.0 + 24.2i)23-s + (1.97 − 3.41i)25-s − 40.3·29-s + (35.0 + 20.2i)31-s + (−33.3 − 57.7i)37-s − 33.6i·41-s + 0.932·43-s + (74.1 − 42.8i)47-s + (−22.2 + 38.6i)53-s − 46.2i·55-s + ⋯ |
| L(s) = 1 | + (−0.931 + 0.537i)5-s + (−0.390 + 0.676i)11-s − 1.61i·13-s + (−0.279 − 0.161i)17-s + (−0.330 + 0.190i)19-s + (0.609 + 1.05i)23-s + (0.0788 − 0.136i)25-s − 1.39·29-s + (1.13 + 0.653i)31-s + (−0.900 − 1.55i)37-s − 0.820i·41-s + 0.0216·43-s + (1.57 − 0.911i)47-s + (−0.420 + 0.728i)53-s − 0.840i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.209497455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209497455\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (4.65 - 2.68i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (4.29 - 7.44i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 21.0iT - 169T^{2} \) |
| 17 | \( 1 + (4.75 + 2.74i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (6.27 - 3.62i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-14.0 - 24.2i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 40.3T + 841T^{2} \) |
| 31 | \( 1 + (-35.0 - 20.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (33.3 + 57.7i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 33.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 0.932T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-74.1 + 42.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (22.2 - 38.6i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (55.1 + 31.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-27.7 + 16.0i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-23.8 + 41.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 14.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-121. - 70.1i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-61.1 - 105. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 33.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (31.2 - 18.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 16.2iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.059755390142747291107066954669, −8.111295371278554455645163674914, −7.47710648406667694890222152216, −7.01243422562012853333479454742, −5.72726226351639799696903854374, −5.08783738089846064800456225961, −3.88229588657198573118537506093, −3.25313650428209334624663794733, −2.14181469817116752872774305573, −0.51422667320134399295908825352,
0.67410453183815055571797847099, 2.06237681351871475703748067688, 3.27261482367572828328189716885, 4.34233872011327979173389498630, 4.71938133617049102242942029233, 6.02951537274194139985354812965, 6.74828788954118132960310889136, 7.65694863828606713658207872640, 8.460984311121314039471732081301, 8.913544807474663846573284548270
