| L(s) = 1 | + (1.11 − 0.863i)2-s + (−1.07 + 1.35i)3-s + (0.507 − 1.93i)4-s + (−1.03 − 1.18i)5-s + (−0.0390 + 2.44i)6-s + (3.17 + 0.418i)7-s + (−1.10 − 2.60i)8-s + (−0.668 − 2.92i)9-s + (−2.18 − 0.427i)10-s + (1.01 + 2.05i)11-s + (2.07 + 2.77i)12-s + (−1.01 − 2.99i)13-s + (3.91 − 2.27i)14-s + (2.71 − 0.127i)15-s + (−3.48 − 1.96i)16-s + (1.39 + 1.39i)17-s + ⋯ |
| L(s) = 1 | + (0.791 − 0.610i)2-s + (−0.623 + 0.781i)3-s + (0.253 − 0.967i)4-s + (−0.463 − 0.528i)5-s + (−0.0159 + 0.999i)6-s + (1.20 + 0.157i)7-s + (−0.389 − 0.920i)8-s + (−0.222 − 0.974i)9-s + (−0.689 − 0.135i)10-s + (0.305 + 0.619i)11-s + (0.598 + 0.801i)12-s + (−0.281 − 0.829i)13-s + (1.04 − 0.607i)14-s + (0.701 − 0.0329i)15-s + (−0.871 − 0.490i)16-s + (0.337 + 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44693 - 1.14778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44693 - 1.14778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.11 + 0.863i)T \) |
| 3 | \( 1 + (1.07 - 1.35i)T \) |
| good | 5 | \( 1 + (1.03 + 1.18i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-3.17 - 0.418i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-1.01 - 2.05i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.01 + 2.99i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-1.39 - 1.39i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.37 + 6.90i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.00698 + 0.0530i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-2.66 + 0.174i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-5.37 + 3.10i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.25 + 6.32i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.41 - 10.7i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (8.12 - 4.00i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-8.22 + 2.20i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.48 - 2.22i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (2.67 + 3.05i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.971 - 14.8i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (8.76 + 4.32i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-0.843 - 2.03i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-0.0877 + 0.211i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.43 - 9.08i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-12.5 - 11.0i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (12.7 - 5.26i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-2.73 - 1.57i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.75059894899462710926287692265, −9.946272104407029597482591731275, −9.032512371898104338882489779643, −7.930673501911002047237581057733, −6.62330430578985738572147700220, −5.44736585253925992787536963762, −4.74852902769565121198221924068, −4.21916899092796598345809153310, −2.75493839657303731859714752619, −0.970506985605089481898954558457,
1.73727111060301059876313523492, 3.32276409647103648214989558626, 4.56144093873604970017286215075, 5.44183443464618775833077468220, 6.40569235263259717588681633544, 7.25063886325556781958397492571, 7.891686869398345106812979637682, 8.657809338461523697280207209766, 10.42306104207998995689450388993, 11.31582207697263731988821507829
