| L(s) = 1 | + (1.54 − 0.454i)2-s + (−0.654 − 0.755i)3-s + (0.506 − 0.325i)4-s + (0.0749 + 0.521i)5-s + (−1.35 − 0.872i)6-s + (0.200 − 0.439i)7-s + (−1.47 + 1.70i)8-s + (−0.142 + 0.989i)9-s + (0.353 + 0.772i)10-s + (−3.21 − 0.943i)11-s + (−0.577 − 0.169i)12-s + (0.853 + 1.86i)13-s + (0.111 − 0.772i)14-s + (0.344 − 0.398i)15-s + (−2.01 + 4.40i)16-s + (−0.919 − 0.590i)17-s + ⋯ |
| L(s) = 1 | + (1.09 − 0.321i)2-s + (−0.378 − 0.436i)3-s + (0.253 − 0.162i)4-s + (0.0335 + 0.233i)5-s + (−0.554 − 0.356i)6-s + (0.0759 − 0.166i)7-s + (−0.522 + 0.602i)8-s + (−0.0474 + 0.329i)9-s + (0.111 + 0.244i)10-s + (−0.968 − 0.284i)11-s + (−0.166 − 0.0489i)12-s + (0.236 + 0.518i)13-s + (0.0296 − 0.206i)14-s + (0.0890 − 0.102i)15-s + (−0.502 + 1.10i)16-s + (−0.223 − 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20569 - 0.302335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20569 - 0.302335i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (1.81 + 4.43i)T \) |
| good | 2 | \( 1 + (-1.54 + 0.454i)T + (1.68 - 1.08i)T^{2} \) |
| 5 | \( 1 + (-0.0749 - 0.521i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.200 + 0.439i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (3.21 + 0.943i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.853 - 1.86i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.919 + 0.590i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-6.19 + 3.98i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (3.03 + 1.94i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (3.65 - 4.21i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.491 + 3.42i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.47 - 10.2i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (5.05 + 5.83i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + (3.06 - 6.71i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (0.390 + 0.854i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (3.75 - 4.33i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (12.3 - 3.62i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (5.99 - 1.76i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.00713 - 0.00458i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-4.29 - 9.40i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.861 + 5.99i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (1.84 + 2.13i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.61 + 18.1i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−14.23432630183480271372951477654, −13.53490981853323164927861101674, −12.62323354203579353470263275211, −11.58057261465295193777320355147, −10.66865575599870260615199888474, −8.856903819043072289275027763916, −7.31735970429080944968569188573, −5.83953416083387693711026092355, −4.63863833419681775585097057188, −2.82825597089822564979806938450,
3.52415423486417004343210322678, 5.08046347029866946072625794704, 5.79231152361413839361787310080, 7.52309602404587436028351821403, 9.257485098595113167663307937036, 10.41450681127969300118964260460, 11.83638940743638968527939037047, 12.82892719409696373130837377377, 13.72186822864152902139386639478, 14.91976514638015796662435516045
