L(s) = 1 | + (−0.876 − 0.876i)2-s + (−1.92 + 1.11i)3-s − 0.463i·4-s + (2.51 + 0.674i)5-s + (2.66 + 0.713i)6-s + (−2.15 + 2.15i)8-s + (0.975 − 1.69i)9-s + (−1.61 − 2.79i)10-s + (1.36 + 0.365i)11-s + (0.515 + 0.893i)12-s + (−0.445 + 3.57i)13-s + (−5.60 + 1.50i)15-s + 2.85·16-s − 2.82·17-s + (−2.33 + 0.626i)18-s + (−1.61 − 6.04i)19-s + ⋯ |
L(s) = 1 | + (−0.619 − 0.619i)2-s + (−1.11 + 0.642i)3-s − 0.231i·4-s + (1.12 + 0.301i)5-s + (1.08 + 0.291i)6-s + (−0.763 + 0.763i)8-s + (0.325 − 0.563i)9-s + (−0.510 − 0.884i)10-s + (0.411 + 0.110i)11-s + (0.148 + 0.257i)12-s + (−0.123 + 0.992i)13-s + (−1.44 + 0.387i)15-s + 0.714·16-s − 0.685·17-s + (−0.550 + 0.147i)18-s + (−0.371 − 1.38i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.422614 + 0.359378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.422614 + 0.359378i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (0.445 - 3.57i)T \) |
good | 2 | \( 1 + (0.876 + 0.876i)T + 2iT^{2} \) |
| 3 | \( 1 + (1.92 - 1.11i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.51 - 0.674i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.36 - 0.365i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + (1.61 + 6.04i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 1.01iT - 23T^{2} \) |
| 29 | \( 1 + (2.66 - 4.61i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.73 - 6.46i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (6.50 - 6.50i)T - 37iT^{2} \) |
| 41 | \( 1 + (-2.51 - 9.40i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.850 + 0.490i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.594 - 2.21i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.52 - 4.38i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.39 - 5.39i)T + 59iT^{2} \) |
| 61 | \( 1 + (-6.75 - 3.89i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.38 - 12.6i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.97 + 11.0i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (8.43 - 2.26i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (2.78 + 4.82i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.445 + 0.445i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.108 + 0.108i)T + 89iT^{2} \) |
| 97 | \( 1 + (-2.87 - 0.771i)T + (84.0 + 48.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.63512251767908671002618335713, −10.11399132737692504723530310479, −9.298371168666533445476769819501, −8.760816236352604049136785388456, −6.83105349197503719056327925884, −6.26706000323094112457097130232, −5.28979473416648627164429404484, −4.52889979550355051888437500504, −2.67561913022065318766577842560, −1.49798748475690753036292985096,
0.42794084375390620616076714501, 2.02997927200871971899396606909, 3.81625317692945496243961227179, 5.49619548515756042002283029696, 5.96398979366731097536783122801, 6.71582027329031096740734447696, 7.65275291845966834994596586334, 8.556889196032203315473964732555, 9.472645628124974566102377630761, 10.23377227005196650503892709683