L(s) = 1 | + (2.77 − 1.13i)3-s + (−6.28 − 6.28i)5-s + 1.64i·7-s + (6.43 − 6.29i)9-s + (−4.75 − 4.75i)11-s + (−9.35 − 9.35i)13-s + (−24.5 − 10.3i)15-s − 11.4i·17-s + (8.58 + 8.58i)19-s + (1.86 + 4.57i)21-s + 16.2·23-s + 54.0i·25-s + (10.7 − 24.7i)27-s + (10.7 − 10.7i)29-s − 6.35·31-s + ⋯ |
L(s) = 1 | + (0.926 − 0.377i)3-s + (−1.25 − 1.25i)5-s + 0.235i·7-s + (0.715 − 0.699i)9-s + (−0.432 − 0.432i)11-s + (−0.719 − 0.719i)13-s + (−1.63 − 0.689i)15-s − 0.675i·17-s + (0.451 + 0.451i)19-s + (0.0887 + 0.217i)21-s + 0.706·23-s + 2.16i·25-s + (0.398 − 0.917i)27-s + (0.370 − 0.370i)29-s − 0.204·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.809989 - 1.17918i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.809989 - 1.17918i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.77 + 1.13i)T \) |
good | 5 | \( 1 + (6.28 + 6.28i)T + 25iT^{2} \) |
| 7 | \( 1 - 1.64iT - 49T^{2} \) |
| 11 | \( 1 + (4.75 + 4.75i)T + 121iT^{2} \) |
| 13 | \( 1 + (9.35 + 9.35i)T + 169iT^{2} \) |
| 17 | \( 1 + 11.4iT - 289T^{2} \) |
| 19 | \( 1 + (-8.58 - 8.58i)T + 361iT^{2} \) |
| 23 | \( 1 - 16.2T + 529T^{2} \) |
| 29 | \( 1 + (-10.7 + 10.7i)T - 841iT^{2} \) |
| 31 | \( 1 + 6.35T + 961T^{2} \) |
| 37 | \( 1 + (-27.2 + 27.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 1.98T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-19.4 + 19.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 74.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (4.00 + 4.00i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (27.9 + 27.9i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-39.2 - 39.2i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-68.6 - 68.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 40.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 17.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (75.1 - 75.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 78.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 38.8T + 9.40e3T^{2} \) |
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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
−12.35200433688280270891122701349, −11.25997191035437021427808419970, −9.697287748425241268706501755949, −8.785978708960119510935578083426, −7.971776067079739265191179480164, −7.32929162337966567471078079600, −5.39514927328910254632705294268, −4.19622509605699924584103702617, −2.88190735855970880694419593674, −0.76972115015039227886187311764,
2.55059706610198536308316647099, 3.64130813587629485274942611988, 4.68861510215831280088986561437, 6.88249771202581406932953121639, 7.46647093766185320140497559318, 8.444374866158069072160267669451, 9.741998713477262614370464126292, 10.61852317315182258386916347078, 11.46326081684758548060338438502, 12.59632046666483497702347177804