L(s) = 1 | + (−0.352 − 0.852i)5-s + (3.43 + 3.43i)7-s + (−1.44 − 3.49i)11-s + (−0.258 − 0.107i)13-s + 5.30·17-s + (−2.72 + 6.57i)19-s + (2.23 + 2.23i)23-s + (2.93 − 2.93i)25-s + (3.16 + 1.31i)29-s − 3.46i·31-s + (1.71 − 4.13i)35-s + (1.27 − 0.528i)37-s + (−5.28 + 5.28i)41-s + (−2.46 + 1.02i)43-s + 0.423i·47-s + ⋯ |
L(s) = 1 | + (−0.157 − 0.381i)5-s + (1.29 + 1.29i)7-s + (−0.436 − 1.05i)11-s + (−0.0717 − 0.0297i)13-s + 1.28·17-s + (−0.625 + 1.50i)19-s + (0.466 + 0.466i)23-s + (0.586 − 0.586i)25-s + (0.588 + 0.243i)29-s − 0.622i·31-s + (0.289 − 0.699i)35-s + (0.209 − 0.0868i)37-s + (−0.824 + 0.824i)41-s + (−0.376 + 0.155i)43-s + 0.0618i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.838715736\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.838715736\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.352 + 0.852i)T + (-3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (-3.43 - 3.43i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.44 + 3.49i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (0.258 + 0.107i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 + (2.72 - 6.57i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.23 - 2.23i)T + 23iT^{2} \) |
| 29 | \( 1 + (-3.16 - 1.31i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-1.27 + 0.528i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (5.28 - 5.28i)T - 41iT^{2} \) |
| 43 | \( 1 + (2.46 - 1.02i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 0.423iT - 47T^{2} \) |
| 53 | \( 1 + (-12.5 + 5.20i)T + (37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (5.24 - 2.17i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.0138 + 0.0333i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-9.82 - 4.06i)T + (47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (-4.64 + 4.64i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.96 - 3.96i)T + 73iT^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + (0.867 + 0.359i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (4.82 + 4.82i)T + 89iT^{2} \) |
| 97 | \( 1 - 8.78T + 97T^{2} \) |
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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
−9.849347525991164358790276514848, −8.730443184413039164510777889294, −8.269108694536758183001753558804, −7.77150224262662815006331932980, −6.25498598654117530945721644433, −5.48812830751214387335600827340, −4.93874701138737132733483034394, −3.62901009892544325090067111671, −2.47515376990035018737265698437, −1.24865146074163315959227093352,
0.990664286366145312849589442730, 2.32432543964116659718234860413, 3.62773031725392868894850156194, 4.71192347161724010678732820392, 5.11644758432640600558229431171, 6.77907998550907514769737538150, 7.23753919497025204922959961390, 7.950355132181693571774246249024, 8.827384864018473517967959943246, 9.998794874547604982961687402861