| L(s) = 1 | + (−1.03 − 0.0904i)2-s + (1.64 − 0.538i)3-s + (−0.909 − 0.160i)4-s + (1.76 − 1.36i)5-s + (−1.75 + 0.407i)6-s + (−2.38 − 1.66i)7-s + (2.92 + 0.785i)8-s + (2.41 − 1.77i)9-s + (−1.95 + 1.25i)10-s + (−0.357 + 0.981i)11-s + (−1.58 + 0.225i)12-s + (−0.102 − 1.17i)13-s + (2.31 + 1.94i)14-s + (2.17 − 3.20i)15-s + (−1.22 − 0.444i)16-s + (5.46 − 1.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.730 − 0.0639i)2-s + (0.950 − 0.310i)3-s + (−0.454 − 0.0801i)4-s + (0.790 − 0.612i)5-s + (−0.714 + 0.166i)6-s + (−0.900 − 0.630i)7-s + (1.03 + 0.277i)8-s + (0.806 − 0.591i)9-s + (−0.616 + 0.397i)10-s + (−0.107 + 0.295i)11-s + (−0.457 + 0.0651i)12-s + (−0.0285 − 0.326i)13-s + (0.617 + 0.518i)14-s + (0.560 − 0.827i)15-s + (−0.305 − 0.111i)16-s + (1.32 − 0.355i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.840355 - 0.449385i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.840355 - 0.449385i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.64 + 0.538i)T \) |
| 5 | \( 1 + (-1.76 + 1.36i)T \) |
| good | 2 | \( 1 + (1.03 + 0.0904i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (2.38 + 1.66i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.357 - 0.981i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.102 + 1.17i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-5.46 + 1.46i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3.77 - 2.17i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.49 - 3.55i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (5.99 - 5.02i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.88 - 10.6i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.84 - 6.89i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.34 - 2.79i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.986 + 2.11i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (0.209 - 0.298i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.16 + 1.16i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.77 + 3.55i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.654 + 3.70i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.570 + 0.0498i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-2.83 - 1.63i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.56 + 13.2i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (2.43 + 2.89i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.416 - 4.75i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (1.62 + 2.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.8 - 5.05i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.12584739580276159932828609255, −12.52821070896558303071967051382, −10.36657470429926328137449402001, −9.805451448805300263373521771894, −9.008238398437937337381584486547, −8.021696666116300365129516839320, −6.88578762846363831042457262030, −5.15368259604981218562270418381, −3.48578958470407191116051242277, −1.41367360153948431061409535629,
2.41762101524913293050598823299, 3.90171177071693557975532545256, 5.75348674494336429831319776470, 7.21148262314934771118975097654, 8.395110107188231945070735260764, 9.401264308835733961738595818458, 9.814510931626059500077096183262, 10.87750226147169640664295879197, 12.81710646700633907035838515059, 13.35842422982689484733608900328
