| L(s) = 1 | + (−0.654 + 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−1.61 − 3.53i)5-s + (0.142 − 0.989i)6-s + (−3.99 + 1.17i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (3.72 + 1.09i)10-s + (−2.96 − 3.42i)11-s + (0.654 + 0.755i)12-s + (3.13 + 0.919i)13-s + (1.72 − 3.78i)14-s + (3.26 + 2.10i)15-s + (−0.959 + 0.281i)16-s + (0.260 − 1.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.721 − 1.58i)5-s + (0.0580 − 0.404i)6-s + (−1.50 + 0.443i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (1.17 + 0.346i)10-s + (−0.894 − 1.03i)11-s + (0.189 + 0.218i)12-s + (0.868 + 0.254i)13-s + (0.462 − 1.01i)14-s + (0.844 + 0.542i)15-s + (−0.239 + 0.0704i)16-s + (0.0631 − 0.438i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.165778 - 0.240778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.165778 - 0.240778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (2.28 - 4.21i)T \) |
| good | 5 | \( 1 + (1.61 + 3.53i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (3.99 - 1.17i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (2.96 + 3.42i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.13 - 0.919i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.260 + 1.81i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.194 + 1.35i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.0798 - 0.555i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (3.90 + 2.50i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.04 + 2.29i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (2.54 + 5.56i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-8.28 + 5.32i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 + (10.1 - 2.98i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-6.86 - 2.01i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.73 + 1.76i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.11 + 4.75i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-4.22 + 4.87i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.75 + 12.2i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (8.09 + 2.37i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.156 - 0.342i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-12.7 + 8.22i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.84 - 17.1i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−12.89058700999390160614011658211, −11.91537949550447083869699917230, −10.77449205439342677119551259362, −9.363566809292221826844103420654, −8.877177663877024821242207252303, −7.68113844721072914269821370069, −6.08840828342920896233376312486, −5.29038655111059805249455139383, −3.71165325706880468394763523266, −0.35217438654691582087542550619,
2.76247369260552375834314944951, 3.90856011211871057868797474926, 6.26768665191951009595194230546, 7.06193696013240410393145424056, 8.017723961279178555268817994232, 9.892940593032381709414526611713, 10.45500464611209806693496389152, 11.22854014276878936101528294283, 12.48982307219212807652670968403, 13.09506029390124506021882537665
