L(s) = 1 | + (2.69 − 0.856i)2-s + (6.53 − 4.61i)4-s − 10.3i·5-s + (−16.6 + 8.15i)7-s + (13.6 − 18.0i)8-s + (−8.83 − 27.7i)10-s − 18.0i·11-s − 49.0i·13-s + (−37.8 + 36.2i)14-s + (21.3 − 60.3i)16-s − 46.6i·17-s − 48.8·19-s + (−47.6 − 67.3i)20-s + (−15.4 − 48.6i)22-s + 33.7i·23-s + ⋯ |
L(s) = 1 | + (0.952 − 0.302i)2-s + (0.816 − 0.577i)4-s − 0.921i·5-s + (−0.897 + 0.440i)7-s + (0.603 − 0.797i)8-s + (−0.279 − 0.878i)10-s − 0.494i·11-s − 1.04i·13-s + (−0.722 + 0.691i)14-s + (0.332 − 0.942i)16-s − 0.666i·17-s − 0.589·19-s + (−0.532 − 0.752i)20-s + (−0.149 − 0.471i)22-s + 0.306i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.635511241\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.635511241\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.69 + 0.856i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (16.6 - 8.15i)T \) |
good | 5 | \( 1 + 10.3iT - 125T^{2} \) |
| 11 | \( 1 + 18.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 49.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 46.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 33.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 48.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 409. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 470. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 548.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 203.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 717.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 493. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 240. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 995. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 790. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 214. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 885.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 67.3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 934. iT - 9.12e5T^{2} \) |
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
−11.53087117628818612416377963465, −10.46299691353386677164596981529, −9.486046200266157042329065146180, −8.431756624996084580608167032433, −7.03207954066678047274236107427, −5.84523237650363436789081163578, −5.12924899231048358129990451278, −3.75410032555723152710774782009, −2.59268138346202208652667003505, −0.74575324925323944111373416928,
2.22534733679483035976098136585, 3.49930011651324140427855874216, 4.44624243907979371323296035420, 6.05678467631100468926205758362, 6.76403019563839163761942773407, 7.49355077892152028173021617359, 8.995178945757699191730531670831, 10.33687396386505735648996873303, 10.97154963589589126677196430039, 12.14064962210040533022720724007