| L(s) = 1 | + (1.09 − 0.890i)2-s + (0.755 + 0.654i)3-s + (0.415 − 1.95i)4-s + (−3.04 − 0.437i)5-s + (1.41 + 0.0468i)6-s + (−0.849 − 1.86i)7-s + (−1.28 − 2.51i)8-s + (0.142 + 0.989i)9-s + (−3.73 + 2.23i)10-s + (−0.297 − 1.01i)11-s + (1.59 − 1.20i)12-s + (−5.65 − 2.58i)13-s + (−2.58 − 1.28i)14-s + (−2.01 − 2.32i)15-s + (−3.65 − 1.62i)16-s + (4.33 − 2.78i)17-s + ⋯ |
| L(s) = 1 | + (0.777 − 0.629i)2-s + (0.436 + 0.378i)3-s + (0.207 − 0.978i)4-s + (−1.36 − 0.195i)5-s + (0.577 + 0.0191i)6-s + (−0.321 − 0.703i)7-s + (−0.454 − 0.890i)8-s + (0.0474 + 0.329i)9-s + (−1.18 + 0.705i)10-s + (−0.0896 − 0.305i)11-s + (0.460 − 0.348i)12-s + (−1.56 − 0.715i)13-s + (−0.692 − 0.344i)14-s + (−0.520 − 0.600i)15-s + (−0.913 − 0.406i)16-s + (1.05 − 0.675i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.562079 - 1.44895i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.562079 - 1.44895i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.09 + 0.890i)T \) |
| 3 | \( 1 + (-0.755 - 0.654i)T \) |
| 23 | \( 1 + (-3.49 - 3.28i)T \) |
| good | 5 | \( 1 + (3.04 + 0.437i)T + (4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (0.849 + 1.86i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.297 + 1.01i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (5.65 + 2.58i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-4.33 + 2.78i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.28 + 5.11i)T + (-7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.04 - 4.73i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.17 - 1.35i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (2.54 - 0.365i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.885 - 6.16i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (3.44 + 2.98i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 + (-12.3 + 5.65i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.90 + 0.869i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-2.75 + 2.39i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (3.51 - 11.9i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (6.40 + 1.88i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (6.61 + 4.25i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.18 + 11.3i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-4.16 + 0.599i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-8.33 + 9.61i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.988 + 6.87i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.43985781947104366266367996340, −9.882521477583939293162869953979, −8.827336420088221732843236194551, −7.49500536112937284333025027860, −7.11463154055804936165504382163, −5.22651389999398831643545132929, −4.70677245337038415243335349232, −3.44227830879863391166727149145, −2.96634896346262394135486303785, −0.65619235378868417186889720348,
2.47174637773773877316236947727, 3.49476128471145243098815644657, 4.44319038532352677431368454863, 5.59089337933925795003664222317, 6.75894902951087606738355440755, 7.55542748263167660324882719320, 8.043995745373284109405640126688, 9.068266539188702620908239522003, 10.21593143620856595222097148757, 11.71257721773955217123968616422
