| L(s) = 1 | + (1.04 + 2.87i)2-s + (1.70 + 0.300i)3-s + (−4.10 + 3.44i)4-s + (0.191 + 0.160i)5-s + (0.919 + 5.21i)6-s + (−3.85 − 6.67i)7-s + (−3.58 − 2.07i)8-s + (2.81 + 1.02i)9-s + (−0.260 + 0.716i)10-s + (4.20 − 7.27i)11-s + (−8.03 + 4.63i)12-s + (−16.3 + 2.87i)13-s + (15.1 − 18.0i)14-s + (0.277 + 0.330i)15-s + (−1.51 + 8.61i)16-s + (10.7 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (0.523 + 1.43i)2-s + (0.568 + 0.100i)3-s + (−1.02 + 0.860i)4-s + (0.0382 + 0.0320i)5-s + (0.153 + 0.869i)6-s + (−0.550 − 0.953i)7-s + (−0.448 − 0.258i)8-s + (0.313 + 0.114i)9-s + (−0.0260 + 0.0716i)10-s + (0.381 − 0.661i)11-s + (−0.669 + 0.386i)12-s + (−1.25 + 0.221i)13-s + (1.08 − 1.29i)14-s + (0.0185 + 0.0220i)15-s + (−0.0949 + 0.538i)16-s + (0.630 − 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07889 + 1.22918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07889 + 1.22918i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 - 0.300i)T \) |
| 19 | \( 1 + (-18.9 + 0.183i)T \) |
| good | 2 | \( 1 + (-1.04 - 2.87i)T + (-3.06 + 2.57i)T^{2} \) |
| 5 | \( 1 + (-0.191 - 0.160i)T + (4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (3.85 + 6.67i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.20 + 7.27i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (16.3 - 2.87i)T + (158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-10.7 + 3.89i)T + (221. - 185. i)T^{2} \) |
| 23 | \( 1 + (-0.904 + 0.758i)T + (91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (8.37 - 23.0i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (49.2 - 28.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 0.192iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (47.4 + 8.35i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (34.4 + 28.9i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-78.2 - 28.4i)T + (1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-3.39 - 4.04i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-35.4 - 97.3i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (22.3 - 18.7i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-22.5 + 62.0i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-27.8 + 33.2i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-9.20 + 52.1i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-147. - 26.0i)T + (5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-51.2 - 88.7i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (62.9 - 11.0i)T + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-0.285 - 0.784i)T + (-7.20e3 + 6.04e3i)T^{2} \) |
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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
−15.17306169841697812930440172515, −14.13532537701901170611377232190, −13.74040132273895889986100241450, −12.33261987739764643606423014803, −10.41045202740282620522651515338, −9.074712733038645806053559019532, −7.60902838816269494259208831713, −6.85799364219773847079648535291, −5.23348618662287527584763624831, −3.66226997979550217683813267279,
2.16261767334825764169361479926, 3.51769391190645272022126571318, 5.27989976710883930163508806610, 7.43059752315182823387163048338, 9.369179587630288863187645002256, 9.904432767023276147754862543171, 11.59563099991925654771308281795, 12.39339911019286593627020898783, 13.17183175534205485005999784131, 14.44952043903756601917232066607
