| L(s) = 1 | + (−1.55 − 1.30i)2-s + (−1.45 + 0.933i)3-s + (0.364 + 2.06i)4-s + (3.47 + 0.449i)6-s + (0.0652 − 0.370i)7-s + (0.101 − 0.176i)8-s + (1.25 − 2.72i)9-s + (0.272 − 0.0993i)11-s + (−2.46 − 2.67i)12-s + (−0.677 + 0.568i)13-s + (−0.583 + 0.489i)14-s + (3.56 − 1.29i)16-s + (−2.32 − 4.02i)17-s + (−5.49 + 2.59i)18-s + (−1.75 + 3.03i)19-s + ⋯ |
| L(s) = 1 | + (−1.09 − 0.920i)2-s + (−0.842 + 0.539i)3-s + (0.182 + 1.03i)4-s + (1.42 + 0.183i)6-s + (0.0246 − 0.139i)7-s + (0.0359 − 0.0622i)8-s + (0.418 − 0.908i)9-s + (0.0823 − 0.0299i)11-s + (−0.711 − 0.772i)12-s + (−0.187 + 0.157i)13-s + (−0.155 + 0.130i)14-s + (0.890 − 0.323i)16-s + (−0.563 − 0.976i)17-s + (−1.29 + 0.611i)18-s + (−0.401 + 0.695i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.394311 + 0.122648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.394311 + 0.122648i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.45 - 0.933i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (1.55 + 1.30i)T + (0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-0.0652 + 0.370i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.272 + 0.0993i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.677 - 0.568i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.32 + 4.02i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.75 - 3.03i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0948 + 0.537i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.65 + 3.90i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.953 - 5.40i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.47 - 9.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.28 - 6.11i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-9.54 + 3.47i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.09 - 6.23i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + (1.18 + 0.432i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.499 + 2.83i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.78 - 5.69i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.36 - 9.29i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.389 + 0.674i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.29 - 4.44i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.88 - 5.77i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (3.11 - 5.38i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.1 - 4.79i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.52234951230043697248086287750, −9.840348990206574738996116196091, −9.226106606382658666032147561194, −8.307348021549343635640449966776, −7.18903475470814063319667309335, −6.11605379437881612844838642962, −5.03648770069329825986481605031, −3.93175024733546852305157035703, −2.58850330367260435836384117536, −1.06478601189250422284587249680,
0.43064993584842492855702380591, 2.04657489386767476428695165647, 4.07096316557271757081150308469, 5.47857956746317986793217944165, 6.15437056767203770731245203560, 7.05482091456594307084002682869, 7.62556774870938086356923744265, 8.607069669557467518372935749060, 9.316865280991189701768725136864, 10.43037531847494923503517003438
