| L(s) = 1 | + (−0.100 − 1.41i)2-s + (0.741 + 0.161i)3-s + (−1.97 + 0.284i)4-s + (−2.52 + 4.31i)5-s + (0.152 − 1.06i)6-s + (1.17 − 2.14i)7-s + (0.601 + 2.76i)8-s + (−7.66 − 3.49i)9-s + (6.34 + 3.13i)10-s + (−7.11 − 8.21i)11-s + (−1.51 − 0.108i)12-s + (−9.90 − 18.1i)13-s + (−3.14 − 1.43i)14-s + (−2.57 + 2.79i)15-s + (3.83 − 1.12i)16-s + (−4.26 − 5.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 − 0.705i)2-s + (0.247 + 0.0537i)3-s + (−0.494 + 0.0711i)4-s + (−0.505 + 0.862i)5-s + (0.0254 − 0.177i)6-s + (0.167 − 0.306i)7-s + (0.0751 + 0.345i)8-s + (−0.851 − 0.388i)9-s + (0.634 + 0.313i)10-s + (−0.646 − 0.746i)11-s + (−0.126 − 0.00902i)12-s + (−0.762 − 1.39i)13-s + (−0.224 − 0.102i)14-s + (−0.171 + 0.186i)15-s + (0.239 − 0.0704i)16-s + (−0.251 − 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0328i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00720037 + 0.438449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00720037 + 0.438449i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.100 + 1.41i)T \) |
| 5 | \( 1 + (2.52 - 4.31i)T \) |
| 23 | \( 1 + (-9.37 - 21.0i)T \) |
| good | 3 | \( 1 + (-0.741 - 0.161i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 2.14i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (7.11 + 8.21i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (9.90 + 18.1i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (4.26 + 5.70i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (22.8 - 3.28i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (-44.1 - 6.35i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (10.2 + 6.57i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (-15.8 - 42.4i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (5.99 + 13.1i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (30.0 + 6.53i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (26.8 - 26.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (62.2 + 33.9i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-30.4 + 103. i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (40.9 + 26.3i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-1.26 - 17.6i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (-41.1 + 47.5i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (39.7 - 53.0i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (-4.29 + 14.6i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-57.8 + 21.5i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (-57.3 - 89.1i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-57.3 - 21.3i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−11.29630007685127682737651261180, −10.70008071764168230161668901088, −9.812927583506456369269009213775, −8.411556512596445043202008635446, −7.81913635004131631968819991223, −6.35275067800031600401803942141, −4.98825992781473206091097556814, −3.39839779171012709345153687532, −2.71419197169023610194271843872, −0.21978430666023449441407111736,
2.26962646963308025472162349157, 4.37148806241410702617730713227, 5.03912354961139393767038324418, 6.47046004949505529311027110615, 7.62071726643833407051624390258, 8.559221052640909066012023601818, 9.087635802332222664697271963393, 10.43994746245525188207789002499, 11.70431857016251260104518502808, 12.53126069391507717521471334207
