| L(s) = 1 | + (0.162 + 1.40i)2-s + (−0.755 + 0.654i)3-s + (−1.94 + 0.456i)4-s + (−2.88 + 0.414i)5-s + (−1.04 − 0.955i)6-s + (−2.13 + 4.68i)7-s + (−0.958 − 2.66i)8-s + (0.142 − 0.989i)9-s + (−1.05 − 3.98i)10-s + (0.715 − 2.43i)11-s + (1.17 − 1.62i)12-s + (1.25 − 0.572i)13-s + (−6.92 − 2.24i)14-s + (1.90 − 2.20i)15-s + (3.58 − 1.77i)16-s + (5.83 + 3.75i)17-s + ⋯ |
| L(s) = 1 | + (0.114 + 0.993i)2-s + (−0.436 + 0.378i)3-s + (−0.973 + 0.228i)4-s + (−1.28 + 0.185i)5-s + (−0.425 − 0.389i)6-s + (−0.807 + 1.76i)7-s + (−0.338 − 0.940i)8-s + (0.0474 − 0.329i)9-s + (−0.332 − 1.25i)10-s + (0.215 − 0.734i)11-s + (0.338 − 0.467i)12-s + (0.347 − 0.158i)13-s + (−1.85 − 0.599i)14-s + (0.492 − 0.568i)15-s + (0.895 − 0.444i)16-s + (1.41 + 0.909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0514590 - 0.0310109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0514590 - 0.0310109i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.162 - 1.40i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (4.40 + 1.90i)T \) |
| good | 5 | \( 1 + (2.88 - 0.414i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (2.13 - 4.68i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.715 + 2.43i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.25 + 0.572i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.83 - 3.75i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (2.30 + 3.58i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.59 + 2.47i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (2.10 - 2.42i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (4.72 + 0.680i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.63 + 11.3i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (8.98 - 7.78i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 3.21T + 47T^{2} \) |
| 53 | \( 1 + (2.87 + 1.31i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (2.75 - 1.25i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.30 - 1.13i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.56 - 8.74i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.77 + 1.10i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (1.93 - 1.24i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.06 - 2.32i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-7.37 - 1.05i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-1.62 - 1.87i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.91 + 13.3i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.81652312418399066615183736765, −10.57847582712246082053421992132, −9.523925172718736794355331629033, −8.554960813627705511346806882321, −8.179073354976278331270680048956, −6.80523325865171902560180711316, −5.99113506395067721694996904669, −5.33297076755066185634894350189, −3.94076912424145701342077002511, −3.19201157337377537830208916321,
0.03946768697758062418361935023, 1.31931865839248575157575451534, 3.45283583482522596025926858496, 3.95294667863382695718267248662, 4.97452322866795553090048784855, 6.49128379739661432000229732398, 7.52883226482897628632050234152, 8.073592279178627046443161958083, 9.601312576803317603427892599693, 10.21264562477278989193981274913
