| L(s) = 1 | + (−0.178 − 1.33i)3-s + (−1.62 − 0.718i)5-s + (−0.315 + 0.678i)7-s + (1.13 − 0.307i)9-s + (1.52 − 1.95i)11-s + (1.00 − 1.96i)13-s + (−0.672 + 2.30i)15-s + (−5.09 − 0.386i)17-s + (1.04 − 1.57i)19-s + (0.964 + 0.301i)21-s + (−4.90 − 1.94i)23-s + (−1.23 − 1.36i)25-s + (−2.18 − 5.20i)27-s + (−2.38 + 0.947i)29-s + (−2.94 + 0.679i)31-s + ⋯ |
| L(s) = 1 | + (−0.103 − 0.772i)3-s + (−0.726 − 0.321i)5-s + (−0.119 + 0.256i)7-s + (0.378 − 0.102i)9-s + (0.458 − 0.588i)11-s + (0.278 − 0.544i)13-s + (−0.173 + 0.595i)15-s + (−1.23 − 0.0937i)17-s + (0.239 − 0.360i)19-s + (0.210 + 0.0657i)21-s + (−1.02 − 0.406i)23-s + (−0.247 − 0.272i)25-s + (−0.420 − 1.00i)27-s + (−0.442 + 0.176i)29-s + (−0.528 + 0.122i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.290062 - 0.871360i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.290062 - 0.871360i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (0.554 + 12.9i)T \) |
| good | 3 | \( 1 + (0.178 + 1.33i)T + (-2.89 + 0.785i)T^{2} \) |
| 5 | \( 1 + (1.62 + 0.718i)T + (3.36 + 3.69i)T^{2} \) |
| 7 | \( 1 + (0.315 - 0.678i)T + (-4.51 - 5.35i)T^{2} \) |
| 11 | \( 1 + (-1.52 + 1.95i)T + (-2.67 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.00 + 1.96i)T + (-7.60 - 10.5i)T^{2} \) |
| 17 | \( 1 + (5.09 + 0.386i)T + (16.8 + 2.56i)T^{2} \) |
| 19 | \( 1 + (-1.04 + 1.57i)T + (-7.35 - 17.5i)T^{2} \) |
| 23 | \( 1 + (4.90 + 1.94i)T + (16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (2.38 - 0.947i)T + (21.0 - 19.9i)T^{2} \) |
| 31 | \( 1 + (2.94 - 0.679i)T + (27.8 - 13.6i)T^{2} \) |
| 37 | \( 1 + (0.619 + 0.168i)T + (31.9 + 18.7i)T^{2} \) |
| 41 | \( 1 + (6.16 - 3.01i)T + (25.2 - 32.3i)T^{2} \) |
| 43 | \( 1 + (-0.510 + 5.37i)T + (-42.2 - 8.08i)T^{2} \) |
| 47 | \( 1 + (0.0873 + 4.61i)T + (-46.9 + 1.77i)T^{2} \) |
| 53 | \( 1 + (-0.991 - 5.76i)T + (-49.9 + 17.7i)T^{2} \) |
| 59 | \( 1 + (-5.11 + 0.387i)T + (58.3 - 8.89i)T^{2} \) |
| 61 | \( 1 + (3.03 - 3.09i)T + (-1.15 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-4.23 + 1.87i)T + (45.0 - 49.5i)T^{2} \) |
| 71 | \( 1 + (-4.23 + 0.482i)T + (69.1 - 15.9i)T^{2} \) |
| 73 | \( 1 + (-10.1 + 7.61i)T + (20.4 - 70.0i)T^{2} \) |
| 79 | \( 1 + (-2.69 + 7.17i)T + (-59.4 - 52.0i)T^{2} \) |
| 83 | \( 1 + (-4.97 + 6.90i)T + (-26.2 - 78.7i)T^{2} \) |
| 89 | \( 1 + (4.27 + 14.6i)T + (-75.0 + 47.8i)T^{2} \) |
| 97 | \( 1 + (9.87 + 2.28i)T + (87.1 + 42.5i)T^{2} \) |
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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
−10.24390386620074256751563695810, −9.086357014606762868260857269057, −8.386040313731112805705683047696, −7.52962950957016618769612215521, −6.66568904426697728426884066079, −5.84138347412438858037939775014, −4.51404290512727271521256446219, −3.56329075258291766407657701640, −2.03586982267943368775781438724, −0.49317958813378934677656235068,
1.91976349271751386701510799933, 3.78656401290258762257147660702, 4.06430896042669064150164577526, 5.23273341526805586047924794325, 6.57743185411132162809314432077, 7.27711335729113682228985133588, 8.257989259491717829434445617346, 9.388895468690849931502333926722, 9.907130717539419148444045126631, 10.94213702567828439348973108398
