| L(s) = 1 | + (0.157 − 0.432i)2-s + (0.888 + 1.48i)3-s + (1.36 + 1.14i)4-s + (0.725 − 4.11i)5-s + (0.783 − 0.149i)6-s + (−1.09 − 2.40i)7-s + (1.51 − 0.871i)8-s + (−1.42 + 2.64i)9-s + (−1.66 − 0.961i)10-s + (−1.36 + 0.240i)11-s + (−0.492 + 3.05i)12-s + (1.43 + 3.94i)13-s + (−1.21 + 0.0968i)14-s + (6.75 − 2.57i)15-s + (0.481 + 2.73i)16-s + (−1.18 + 2.06i)17-s + ⋯ |
| L(s) = 1 | + (0.111 − 0.305i)2-s + (0.512 + 0.858i)3-s + (0.684 + 0.574i)4-s + (0.324 − 1.83i)5-s + (0.319 − 0.0612i)6-s + (−0.415 − 0.909i)7-s + (0.533 − 0.308i)8-s + (−0.474 + 0.880i)9-s + (−0.526 − 0.303i)10-s + (−0.410 + 0.0724i)11-s + (−0.142 + 0.882i)12-s + (0.397 + 1.09i)13-s + (−0.324 + 0.0258i)14-s + (1.74 − 0.664i)15-s + (0.120 + 0.682i)16-s + (−0.288 + 0.499i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61420 - 0.135230i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61420 - 0.135230i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.888 - 1.48i)T \) |
| 7 | \( 1 + (1.09 + 2.40i)T \) |
| good | 2 | \( 1 + (-0.157 + 0.432i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.725 + 4.11i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (1.36 - 0.240i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.43 - 3.94i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.18 - 2.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.72 - 2.72i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.01 - 1.21i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.495 + 1.36i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.18 + 1.41i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.37 + 5.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.07 + 2.21i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.819 - 4.64i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.18 - 1.83i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 11.6iT - 53T^{2} \) |
| 59 | \( 1 + (1.15 - 6.54i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.37 - 1.64i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-9.25 + 3.36i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (9.79 + 5.65i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.69 + 4.44i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.14 + 0.779i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (10.9 + 4.00i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (4.23 + 7.34i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.77 + 1.01i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.80441906166320208764545118635, −11.54785080730122391492805301637, −10.51520642997769433423349293180, −9.554672851933506575118212077330, −8.600560609395047671702893366951, −7.76857405955912013927177180703, −6.10726762863981369390154609099, −4.47429869613565063676352890461, −3.91560939571673136298560855465, −1.94986252411317926917117862073,
2.38897476155400260002639036517, 2.95410343953467303503480125713, 5.75267192967562608122340344430, 6.43571420953456734015430531230, 7.15634167159800749441467521716, 8.279401532055833183129746883503, 9.757932041435275534223528629101, 10.72999825011400527958736195162, 11.46748669130725451982944013911, 12.75195932212777301893549759738
