| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)6-s + (−0.814 + 1.41i)7-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.939 + 0.342i)10-s + (−0.0122 − 0.0211i)11-s + (−0.5 + 0.866i)12-s + (−0.900 + 5.10i)13-s + (1.24 − 1.04i)14-s + (0.766 + 0.642i)15-s + (0.173 + 0.984i)16-s + (3.76 + 1.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (0.0708 − 0.402i)6-s + (−0.307 + 0.532i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.297 + 0.108i)10-s + (−0.00368 − 0.00637i)11-s + (−0.144 + 0.249i)12-s + (−0.249 + 1.41i)13-s + (0.333 − 0.279i)14-s + (0.197 + 0.165i)15-s + (0.0434 + 0.246i)16-s + (0.912 + 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0172 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0172 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.676700 + 0.665135i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.676700 + 0.665135i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (4.12 + 1.40i)T \) |
| good | 7 | \( 1 + (0.814 - 1.41i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0122 + 0.0211i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.900 - 5.10i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.76 - 1.36i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.28 - 3.59i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.96 - 1.08i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.54 - 4.41i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.15T + 37T^{2} \) |
| 41 | \( 1 + (-0.172 - 0.980i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (2.88 - 2.41i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (1.85 - 0.674i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-8.48 - 7.11i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (4.78 + 1.73i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-3.32 - 2.78i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (8.26 - 3.00i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.16 + 4.33i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.30 - 7.42i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.0776 - 0.440i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.37 + 11.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.87 + 10.6i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (5.43 + 1.97i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.81414469517061388987138986937, −9.932946282106454874914738321436, −9.130650392103279949563040074206, −8.784337071583023057870376379464, −7.51077165655063321701587540817, −6.47962739708870535960675232327, −5.43119016780633367813375992903, −4.25762160559826874617065429998, −3.00788096309254235439929439168, −1.71469339078537047454275376230,
0.66289138914219624494340866821, 2.30308174981332939522658490729, 3.49234427241173345748766698889, 5.24630665351654066135972050989, 6.16123899678252301307419529314, 7.08681998982723320992514166544, 7.78956061272158118082042241391, 8.631319512383580674166056918650, 9.744351456599340842105648840277, 10.37954991463162585758550382865
