| L(s) = 1 | + (−1.27 − 1.51i)2-s + (0.476 + 2.70i)3-s + (−0.334 + 1.89i)4-s + (−2.20 + 0.388i)5-s + (3.49 − 4.16i)6-s + (−2.50 + 0.840i)7-s + (−0.130 + 0.0752i)8-s + (−4.26 + 1.55i)9-s + (3.39 + 2.85i)10-s − 2.98·11-s − 5.28·12-s + (1.98 + 1.66i)13-s + (4.46 + 2.73i)14-s + (−2.10 − 5.77i)15-s + (3.89 + 1.41i)16-s + (−0.0795 + 0.218i)17-s + ⋯ |
| L(s) = 1 | + (−0.900 − 1.07i)2-s + (0.275 + 1.56i)3-s + (−0.167 + 0.947i)4-s + (−0.986 + 0.173i)5-s + (1.42 − 1.70i)6-s + (−0.948 + 0.317i)7-s + (−0.0460 + 0.0266i)8-s + (−1.42 + 0.517i)9-s + (1.07 + 0.901i)10-s − 0.901·11-s − 1.52·12-s + (0.551 + 0.462i)13-s + (1.19 + 0.731i)14-s + (−0.543 − 1.49i)15-s + (0.974 + 0.354i)16-s + (−0.0192 + 0.0530i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.235412 + 0.305095i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.235412 + 0.305095i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.50 - 0.840i)T \) |
| 19 | \( 1 + (-4.32 + 0.575i)T \) |
| good | 2 | \( 1 + (1.27 + 1.51i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.476 - 2.70i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (2.20 - 0.388i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 + (-1.98 - 1.66i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0795 - 0.218i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.75 - 4.82i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (7.57 + 1.33i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.284 - 0.493i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (8.35 - 4.82i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.18 + 5.18i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.51 - 2.00i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.685 - 1.88i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.14 + 0.201i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (1.27 + 0.463i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (8.66 - 10.3i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.716 + 0.853i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.39 - 6.58i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-3.45 + 0.609i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-5.24 + 14.4i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.62 - 2.09i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.0865 + 0.490i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.769 - 4.36i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−13.37520223168612131672677192832, −12.02040983981485837449630334656, −11.13766101541031229801489640390, −10.48597549358615829103725835871, −9.420974500319317477273732986897, −8.993436766239675066513423590909, −7.61679388759425494368747610730, −5.46732313270682999755090939190, −3.75652365901930166803420898227, −3.01340798849757142036284957757,
0.50490175143665951789319060456, 3.21219177687499963520460612654, 5.77999977855725068577764647232, 6.96169826475654356384245282825, 7.54360894424326327173669959515, 8.262326214098966871628959367269, 9.316281650957080288043739848408, 10.86528966917069275957359099614, 12.36838738939279908973613344699, 12.85820038051266403298145640635
