| 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 | \( 1 + (-5.77 - 1.01i)T + (91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.75195932212777301893549759738, −11.46748669130725451982944013911, −10.72999825011400527958736195162, −9.757932041435275534223528629101, −8.279401532055833183129746883503, −7.15634167159800749441467521716, −6.43571420953456734015430531230, −5.75267192967562608122340344430, −2.95410343953467303503480125713, −2.38897476155400260002639036517,
1.94986252411317926917117862073, 3.91560939571673136298560855465, 4.47429869613565063676352890461, 6.10726762863981369390154609099, 7.76857405955912013927177180703, 8.600560609395047671702893366951, 9.554672851933506575118212077330, 10.51520642997769433423349293180, 11.54785080730122391492805301637, 12.80441906166320208764545118635
