| L(s) = 1 | + (−0.891 + 0.827i)2-s + (−0.514 + 0.0775i)3-s + (−0.0388 + 0.518i)4-s + (0.948 − 2.41i)5-s + (0.394 − 0.495i)6-s + (−1.91 − 2.39i)8-s + (−2.60 + 0.804i)9-s + (1.15 + 2.93i)10-s + (−3.67 − 1.13i)11-s + (−0.0202 − 0.269i)12-s + (−0.203 − 0.891i)13-s + (−0.300 + 1.31i)15-s + (2.66 + 0.400i)16-s + (−4.07 + 2.77i)17-s + (1.66 − 2.87i)18-s + (−1.24 − 2.15i)19-s + ⋯ |
| L(s) = 1 | + (−0.630 + 0.585i)2-s + (−0.297 + 0.0447i)3-s + (−0.0194 + 0.259i)4-s + (0.423 − 1.08i)5-s + (0.161 − 0.202i)6-s + (−0.675 − 0.847i)8-s + (−0.869 + 0.268i)9-s + (0.364 + 0.929i)10-s + (−1.10 − 0.341i)11-s + (−0.00583 − 0.0779i)12-s + (−0.0564 − 0.247i)13-s + (−0.0776 + 0.340i)15-s + (0.665 + 0.100i)16-s + (−0.988 + 0.674i)17-s + (0.391 − 0.677i)18-s + (−0.285 − 0.494i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.139313 - 0.204280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.139313 - 0.204280i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (0.891 - 0.827i)T + (0.149 - 1.99i)T^{2} \) |
| 3 | \( 1 + (0.514 - 0.0775i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (-0.948 + 2.41i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (3.67 + 1.13i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (0.203 + 0.891i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (4.07 - 2.77i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (1.24 + 2.15i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.26 + 2.22i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (4.93 + 2.37i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-4.93 + 8.54i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.163 - 2.18i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (1.31 + 1.65i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (2.67 - 3.35i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-4.19 + 3.89i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.574 - 7.65i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (3.54 + 9.03i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-1.08 - 14.4i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (3.38 - 5.85i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.32 + 2.56i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (8.33 + 7.73i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-2.95 - 5.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.604 - 2.64i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.929 + 0.286i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−11.18078531161635257273695068339, −10.13947851828205235273589610514, −9.058327252934767817837069482766, −8.421976081060583850248834503001, −7.76276720539061758562728479358, −6.30970862808236279100246767559, −5.53423635775015383620396782647, −4.32902160992121609920534946721, −2.56857524833531686729690878032, −0.20013959891711699151849400137,
2.10793698765450291506784929716, 3.05192982646695486732055981046, 5.06443815205869428457423096012, 6.02112754810169613622541705599, 6.94799099517478694146981685871, 8.270196461489757622481623634476, 9.258789738104719010081463954320, 10.18323767668664212004955424486, 10.77930464218114331551198306566, 11.43691612870526223448385130059
