| L(s) = 1 | + (−1.42 − 0.996i)2-s + (0.949 + 1.44i)3-s + (0.347 + 0.955i)4-s + (−0.893 + 2.04i)5-s + (0.0912 − 3.00i)6-s + (0.429 + 0.920i)7-s + (−0.442 + 1.64i)8-s + (−1.19 + 2.75i)9-s + (3.31 − 2.02i)10-s + (−0.759 − 0.904i)11-s + (−1.05 + 1.41i)12-s + (2.21 + 3.16i)13-s + (0.306 − 1.73i)14-s + (−3.81 + 0.652i)15-s + (3.83 − 3.21i)16-s + (0.144 + 0.540i)17-s + ⋯ |
| L(s) = 1 | + (−1.00 − 0.704i)2-s + (0.548 + 0.836i)3-s + (0.173 + 0.477i)4-s + (−0.399 + 0.916i)5-s + (0.0372 − 1.22i)6-s + (0.162 + 0.347i)7-s + (−0.156 + 0.583i)8-s + (−0.398 + 0.917i)9-s + (1.04 − 0.640i)10-s + (−0.228 − 0.272i)11-s + (−0.304 + 0.407i)12-s + (0.614 + 0.877i)13-s + (0.0818 − 0.464i)14-s + (−0.985 + 0.168i)15-s + (0.957 − 0.803i)16-s + (0.0351 + 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.631237 + 0.302968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.631237 + 0.302968i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.949 - 1.44i)T \) |
| 5 | \( 1 + (0.893 - 2.04i)T \) |
| good | 2 | \( 1 + (1.42 + 0.996i)T + (0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-0.429 - 0.920i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (0.759 + 0.904i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.21 - 3.16i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.144 - 0.540i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.83 + 2.21i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.21 - 1.96i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (1.44 + 8.17i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (7.51 - 2.73i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (5.30 - 1.42i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.46 - 0.964i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.0660 + 0.754i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-11.1 + 5.19i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (5.40 + 5.40i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.20 - 5.20i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-10.5 - 3.83i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.29 - 0.906i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-6.95 - 4.01i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.11 + 1.36i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.62 + 1.16i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.172 + 0.245i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (5.41 + 9.38i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.10 + 0.446i)T + (95.5 + 16.8i)T^{2} \) |
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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
−13.58882797859348555450102223136, −11.66122171580145088124858798346, −11.15564570616634822985724424165, −10.27768973602408382577671131038, −9.292785001332427730893690050711, −8.534229650254686337517634186327, −7.34928035293798103871554019001, −5.47551344175331852103790840863, −3.68160038802314349093828463199, −2.38462692707774874389201006647,
1.03143346717246588143716721614, 3.59392214231536129298438288334, 5.59889833403283475575398010403, 7.17970346694116796925023587256, 7.76559130474402835256250823180, 8.693906086757475359447002140364, 9.403954405498905954863252427680, 10.90003011131118554112198731817, 12.50593735566817146941343710262, 12.84503313097132561002111505543
