| L(s) = 1 | + (−0.926 + 1.06i)2-s + (−1.45 + 0.936i)3-s + (−0.284 − 1.97i)4-s + (1.12 − 0.512i)5-s + (0.348 − 2.42i)6-s + (2.20 + 7.51i)7-s + (2.37 + 1.52i)8-s + (1.24 − 2.72i)9-s + (−0.491 + 1.67i)10-s + (−15.5 + 13.5i)11-s + (2.26 + 2.61i)12-s + (−14.5 − 4.27i)13-s + (−10.0 − 4.60i)14-s + (−1.15 + 1.79i)15-s + (−3.83 + 1.12i)16-s + (−19.6 − 2.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.224 − 0.102i)5-s + (0.0580 − 0.404i)6-s + (0.315 + 1.07i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.0491 + 0.167i)10-s + (−1.41 + 1.22i)11-s + (0.189 + 0.218i)12-s + (−1.12 − 0.328i)13-s + (−0.720 − 0.328i)14-s + (−0.0770 + 0.119i)15-s + (−0.239 + 0.0704i)16-s + (−1.15 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0925480 + 0.586089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0925480 + 0.586089i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.926 - 1.06i)T \) |
| 3 | \( 1 + (1.45 - 0.936i)T \) |
| 23 | \( 1 + (20.9 - 9.49i)T \) |
| good | 5 | \( 1 + (-1.12 + 0.512i)T + (16.3 - 18.8i)T^{2} \) |
| 7 | \( 1 + (-2.20 - 7.51i)T + (-41.2 + 26.4i)T^{2} \) |
| 11 | \( 1 + (15.5 - 13.5i)T + (17.2 - 119. i)T^{2} \) |
| 13 | \( 1 + (14.5 + 4.27i)T + (142. + 91.3i)T^{2} \) |
| 17 | \( 1 + (19.6 + 2.82i)T + (277. + 81.4i)T^{2} \) |
| 19 | \( 1 + (-32.9 + 4.74i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (3.31 - 23.0i)T + (-806. - 236. i)T^{2} \) |
| 31 | \( 1 + (-14.5 - 9.35i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (-30.6 - 13.9i)T + (896. + 1.03e3i)T^{2} \) |
| 41 | \( 1 + (-25.2 - 55.3i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (28.8 + 44.8i)T + (-768. + 1.68e3i)T^{2} \) |
| 47 | \( 1 - 57.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + (27.9 + 95.0i)T + (-2.36e3 + 1.51e3i)T^{2} \) |
| 59 | \( 1 + (-11.8 - 3.48i)T + (2.92e3 + 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-22.3 + 34.8i)T + (-1.54e3 - 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-65.1 - 56.4i)T + (638. + 4.44e3i)T^{2} \) |
| 71 | \( 1 + (-17.9 + 20.7i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-3.45 - 24.0i)T + (-5.11e3 + 1.50e3i)T^{2} \) |
| 79 | \( 1 + (15.9 - 54.2i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-30.1 - 13.7i)T + (4.51e3 + 5.20e3i)T^{2} \) |
| 89 | \( 1 + (2.22 + 3.45i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (103. - 47.4i)T + (6.16e3 - 7.11e3i)T^{2} \) |
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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
−13.36724769275344418511371366616, −12.29421627457921356436554941086, −11.34498677592319477124201695342, −9.977731987519310665525310906581, −9.479633720031751851799570663537, −8.039590828857048825452631973682, −7.06525742448209046380880429061, −5.44461023711110444049380735338, −4.97089159413991887116669649868, −2.33920931461251281805565285346,
0.46214239915212153598133421204, 2.49909682828114588658221414739, 4.36049576759040536817561776570, 5.83470427155650174186408156722, 7.37712854961328168278056384709, 8.065371601268692479731585905608, 9.676003366385817105047267106448, 10.55512180502820518484083082919, 11.29573992292879557566251711005, 12.32093564259246832847652696557
