L(s) = 1 | + (−0.951 − 0.309i)2-s + (1.74 − 2.39i)3-s + (0.809 + 0.587i)4-s + (−2.39 + 1.74i)6-s − 1.83i·7-s + (−0.587 − 0.809i)8-s + (−1.79 − 5.51i)9-s + (−0.566 + 1.74i)11-s + (2.82 − 0.916i)12-s + (2.29 − 0.747i)13-s + (−0.566 + 1.74i)14-s + (0.309 + 0.951i)16-s + (1.63 + 2.25i)17-s + 5.79i·18-s + (−1.35 + 0.982i)19-s + ⋯ |
L(s) = 1 | + (−0.672 − 0.218i)2-s + (1.00 − 1.38i)3-s + (0.404 + 0.293i)4-s + (−0.979 + 0.711i)6-s − 0.692i·7-s + (−0.207 − 0.286i)8-s + (−0.597 − 1.83i)9-s + (−0.170 + 0.525i)11-s + (0.814 − 0.264i)12-s + (0.637 − 0.207i)13-s + (−0.151 + 0.466i)14-s + (0.0772 + 0.237i)16-s + (0.396 + 0.546i)17-s + 1.36i·18-s + (−0.310 + 0.225i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.755042 - 0.976165i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755042 - 0.976165i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.74 + 2.39i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 1.83iT - 7T^{2} \) |
| 11 | \( 1 + (0.566 - 1.74i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.29 + 0.747i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.63 - 2.25i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.35 - 0.982i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (7.38 + 2.39i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.13 - 4.45i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.28 + 3.11i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.25 - 0.406i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 3.34i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 4.30iT - 43T^{2} \) |
| 47 | \( 1 + (1.07 - 1.48i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.83 - 5.27i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.79 - 8.61i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.799 + 2.46i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-5.58 - 7.68i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (0.247 + 0.179i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-14.2 - 4.61i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (2.79 + 2.03i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.74 - 5.15i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.02 + 3.15i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (6.51 - 8.97i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.06933186169827244954547232452, −10.67190681935284569112734162982, −9.797628354695829656025965370117, −8.475852364033278624428088749595, −8.046685657108715893300312970882, −7.08471140027964499000901935144, −6.20611015952871035403748689248, −3.90767165406450634987068540060, −2.52237095166231115692045626419, −1.23893517733004973139220850696,
2.46325660678732980159800034747, 3.67986324567408968705652163656, 5.04276868265040852296441539866, 6.26530871975456040055031251616, 8.016653615523136592274411134738, 8.522131575762855090439461901793, 9.444825495108695974698730578696, 10.08974772498904441362261556378, 11.04418016393780164514355043282, 12.05381411943208985995676764475