| L(s) = 1 | + (1.42 + 1.19i)2-s + (0.697 + 1.58i)3-s + (0.257 + 1.46i)4-s + (−0.939 − 0.342i)5-s + (−0.904 + 3.10i)6-s + (0.603 − 3.42i)7-s + (0.482 − 0.835i)8-s + (−2.02 + 2.21i)9-s + (−0.933 − 1.61i)10-s + (−3.21 + 1.17i)11-s + (−2.13 + 1.42i)12-s + (−2.29 + 1.92i)13-s + (4.97 − 4.17i)14-s + (−0.113 − 1.72i)15-s + (4.47 − 1.63i)16-s + (−0.915 − 1.58i)17-s + ⋯ |
| L(s) = 1 | + (1.01 + 0.848i)2-s + (0.402 + 0.915i)3-s + (0.128 + 0.730i)4-s + (−0.420 − 0.152i)5-s + (−0.369 + 1.26i)6-s + (0.228 − 1.29i)7-s + (0.170 − 0.295i)8-s + (−0.675 + 0.737i)9-s + (−0.295 − 0.511i)10-s + (−0.970 + 0.353i)11-s + (−0.616 + 0.412i)12-s + (−0.635 + 0.533i)13-s + (1.32 − 1.11i)14-s + (−0.0292 − 0.446i)15-s + (1.11 − 0.407i)16-s + (−0.221 − 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38542 + 1.12037i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38542 + 1.12037i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.697 - 1.58i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| good | 2 | \( 1 + (-1.42 - 1.19i)T + (0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-0.603 + 3.42i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (3.21 - 1.17i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.29 - 1.92i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.915 + 1.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.83 + 4.90i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.47 - 8.37i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.71 - 2.28i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.629 - 3.57i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.12 + 7.14i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.49 + 2.93i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.40 - 2.33i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.749 - 4.24i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 6.55T + 53T^{2} \) |
| 59 | \( 1 + (3.06 + 1.11i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.589 + 3.34i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.19 - 2.67i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.35 - 7.54i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.97 + 6.87i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.2 - 9.42i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (11.0 + 9.28i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.78 + 3.09i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.43 + 1.25i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.83286091379350843396784018426, −12.91572017699293275618514795624, −11.38360408855458468429519034103, −10.36757021955056667279223075773, −9.341306996333330734761586986002, −7.68906006977722896390917721478, −7.11795299643825633733779582698, −5.16393392713446487028780968441, −4.60599266912071917276597433778, −3.38709368600828228013185789041,
2.28947588621728108218466795686, 3.17049727872104141093928173710, 4.99893508977059860602341064281, 6.12362590966543785490224407037, 7.892507740975846145379027446411, 8.479144992939686656793144541745, 10.26490218093409998240707600166, 11.50320960318005345546355093741, 12.26384670807053675350093092519, 12.74146713703527583513415125192
