| L(s) = 1 | + (2.45 + 0.433i)2-s + (−1.54 + 0.776i)3-s + (3.97 + 1.44i)4-s + (−0.335 − 0.122i)5-s + (−4.14 + 1.23i)6-s + (0.0666 + 2.64i)7-s + (4.81 + 2.77i)8-s + (1.79 − 2.40i)9-s + (−0.772 − 0.446i)10-s + (−1.17 − 3.23i)11-s + (−7.27 + 0.844i)12-s + (0.362 − 0.995i)13-s + (−0.982 + 6.52i)14-s + (0.615 − 0.0714i)15-s + (4.14 + 3.47i)16-s + (2.91 − 5.04i)17-s + ⋯ |
| L(s) = 1 | + (1.73 + 0.306i)2-s + (−0.893 + 0.448i)3-s + (1.98 + 0.722i)4-s + (−0.150 − 0.0546i)5-s + (−1.69 + 0.504i)6-s + (0.0251 + 0.999i)7-s + (1.70 + 0.982i)8-s + (0.598 − 0.801i)9-s + (−0.244 − 0.141i)10-s + (−0.355 − 0.975i)11-s + (−2.09 + 0.243i)12-s + (0.100 − 0.276i)13-s + (−0.262 + 1.74i)14-s + (0.158 − 0.0184i)15-s + (1.03 + 0.869i)16-s + (0.706 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.01220 + 0.953610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01220 + 0.953610i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.54 - 0.776i)T \) |
| 7 | \( 1 + (-0.0666 - 2.64i)T \) |
| good | 2 | \( 1 + (-2.45 - 0.433i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (0.335 + 0.122i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (1.17 + 3.23i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.362 + 0.995i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.91 + 5.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.53 - 2.04i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.12 - 0.374i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.91 - 7.99i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.29 + 3.54i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 + (4.24 + 1.54i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.24 - 7.08i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-11.5 + 4.19i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-7.91 + 4.56i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.15 - 2.65i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.54 - 12.4i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.801 - 4.54i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (7.24 - 4.18i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.45iT - 73T^{2} \) |
| 79 | \( 1 + (2.32 - 13.2i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (3.80 - 1.38i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (5.79 + 10.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.45 + 0.785i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.57253061034804583743772587969, −11.95332979474526812874712673866, −11.29222128769572450486753579773, −10.11444501694011405165862375565, −8.505992614013757176507578032303, −7.00342245785747725307690866880, −5.78817927820297153462749214344, −5.45019808762823556178320382908, −4.17398657979854456098332603747, −2.90095772562431744009588628893,
1.93029255266169803296511006158, 3.90891261674889844676675680426, 4.70986638718117258699889235615, 5.92595664491615979377075760985, 6.81708284796914401227396734084, 7.77541782205732547560728676559, 10.21803265174750549378862060204, 10.74997381019650542504482318569, 11.85524801523520779554083832208, 12.45340721416273437695911561252
