L(s) = 1 | + (−1.5 − 0.866i)3-s + (−1.24 + 0.717i)5-s + (−1.74 + 6.77i)7-s + (1.5 + 2.59i)9-s + (3 − 5.19i)11-s − 21.3i·13-s + 2.48·15-s + (−7.75 − 4.47i)17-s + (6.25 − 3.61i)19-s + (8.48 − 8.66i)21-s + (−18.7 − 32.4i)23-s + (−11.4 + 19.8i)25-s − 5.19i·27-s − 33.9·29-s + (−38.2 − 22.0i)31-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.288i)3-s + (−0.248 + 0.143i)5-s + (−0.248 + 0.968i)7-s + (0.166 + 0.288i)9-s + (0.272 − 0.472i)11-s − 1.64i·13-s + 0.165·15-s + (−0.456 − 0.263i)17-s + (0.329 − 0.190i)19-s + (0.404 − 0.412i)21-s + (−0.814 − 1.41i)23-s + (−0.458 + 0.794i)25-s − 0.192i·27-s − 1.17·29-s + (−1.23 − 0.711i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.279188 - 0.561843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.279188 - 0.561843i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (1.74 - 6.77i)T \) |
good | 5 | \( 1 + (1.24 - 0.717i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 21.3iT - 169T^{2} \) |
| 17 | \( 1 + (7.75 + 4.47i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.25 + 3.61i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (18.7 + 32.4i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 33.9T + 841T^{2} \) |
| 31 | \( 1 + (38.2 + 22.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-13.9 - 24.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 54.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.48T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-37.2 + 21.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-42.7 + 74.0i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-35.6 - 20.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.02 - 0.594i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-2.19 + 3.80i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 137.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-68.3 - 39.4i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-49.1 - 85.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (18 - 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 10.9iT - 9.40e3T^{2} \) |
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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
−11.14052378801537224038014951444, −10.21027850519015080806706054815, −9.085401670083490569088607922984, −8.151835481077152472907957382424, −7.14497368805639321203491959667, −5.92285178242963670527150351303, −5.35241684172126701457131833665, −3.68878024531006319202674696792, −2.35070845570825123393360283764, −0.30694248599785445817418134248,
1.65233983664443142631883754336, 3.82836960282743377088542992836, 4.39094462496383554117165315713, 5.83396232702134959754649287498, 6.90785231855669327180410414564, 7.65419972440289389774311525835, 9.159062715593132092511158498251, 9.751919295857267957583741724113, 10.85326984065199690400081492160, 11.57492397230277680847894361117