| L(s) = 1 | + (−1.46 + 1.36i)2-s + (0.159 − 0.596i)3-s + (0.271 − 3.99i)4-s + (−4.27 + 2.59i)5-s + (0.580 + 1.08i)6-s + (1.81 − 6.75i)7-s + (5.05 + 6.20i)8-s + (7.46 + 4.30i)9-s + (2.70 − 9.62i)10-s + (1.62 − 0.937i)11-s + (−2.33 − 0.799i)12-s + (15.6 − 15.6i)13-s + (6.57 + 12.3i)14-s + (0.864 + 2.96i)15-s + (−15.8 − 2.16i)16-s + (2.93 − 10.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.730 + 0.682i)2-s + (0.0532 − 0.198i)3-s + (0.0679 − 0.997i)4-s + (−0.854 + 0.519i)5-s + (0.0967 + 0.181i)6-s + (0.259 − 0.965i)7-s + (0.631 + 0.775i)8-s + (0.829 + 0.478i)9-s + (0.270 − 0.962i)10-s + (0.147 − 0.0851i)11-s + (−0.194 − 0.0666i)12-s + (1.20 − 1.20i)13-s + (0.469 + 0.883i)14-s + (0.0576 + 0.197i)15-s + (−0.990 − 0.135i)16-s + (0.172 − 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.990491 + 0.0288450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.990491 + 0.0288450i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.46 - 1.36i)T \) |
| 5 | \( 1 + (4.27 - 2.59i)T \) |
| 7 | \( 1 + (-1.81 + 6.75i)T \) |
| good | 3 | \( 1 + (-0.159 + 0.596i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (-1.62 + 0.937i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-15.6 + 15.6i)T - 169iT^{2} \) |
| 17 | \( 1 + (-2.93 + 10.9i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-24.9 - 14.4i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-1.74 - 6.52i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 22.7iT - 841T^{2} \) |
| 31 | \( 1 + (4.22 + 7.31i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (7.17 + 26.7i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 23.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (41.7 - 41.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (20.8 + 77.8i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-5.90 + 22.0i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-5.82 + 3.36i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-49.7 - 28.7i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (26.5 - 99.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 77.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-18.5 - 4.97i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-66.7 + 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-60.1 - 60.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (38.5 - 66.8i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (55.0 + 55.0i)T + 9.40e3iT^{2} \) |
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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
−13.20399834745679219315357333791, −11.58401495864085368372806870031, −10.66932243938252150748913263692, −9.973780338951140939276967240732, −8.331284506321637934634562908180, −7.59071467503008004677374210936, −6.89020787472337187976598373822, −5.29338367809370178152397901571, −3.66710740472300897085909518884, −1.05351527303502169845781250493,
1.41423485293749599486766388463, 3.47264938359254453924129074930, 4.60967634712247116179881506602, 6.64576917091228673018372257805, 7.977667862267797106369686589882, 8.913525415417631951257225229237, 9.572251821058284577453721354368, 11.07554888338540316138654270759, 11.79404439246980971891693304660, 12.50000074125173658880086294513
