| L(s) = 1 | + (1.74 − 0.976i)2-s + (2.40 − 1.79i)3-s + (2.09 − 3.40i)4-s + (−7.02 + 2.55i)5-s + (2.44 − 5.47i)6-s + (8.30 − 1.46i)7-s + (0.325 − 7.99i)8-s + (2.57 − 8.62i)9-s + (−9.76 + 11.3i)10-s + (−3.00 + 8.26i)11-s + (−1.07 − 11.9i)12-s + (5.39 + 4.53i)13-s + (13.0 − 10.6i)14-s + (−12.3 + 18.7i)15-s + (−7.23 − 14.2i)16-s + (−10.7 + 18.6i)17-s + ⋯ |
| L(s) = 1 | + (0.872 − 0.488i)2-s + (0.801 − 0.597i)3-s + (0.523 − 0.852i)4-s + (−1.40 + 0.511i)5-s + (0.408 − 0.912i)6-s + (1.18 − 0.209i)7-s + (0.0406 − 0.999i)8-s + (0.286 − 0.958i)9-s + (−0.976 + 1.13i)10-s + (−0.273 + 0.751i)11-s + (−0.0894 − 0.995i)12-s + (0.415 + 0.348i)13-s + (0.933 − 0.762i)14-s + (−0.821 + 1.25i)15-s + (−0.452 − 0.891i)16-s + (−0.634 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.98396 - 1.34234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98396 - 1.34234i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.74 + 0.976i)T \) |
| 3 | \( 1 + (-2.40 + 1.79i)T \) |
| good | 5 | \( 1 + (7.02 - 2.55i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-8.30 + 1.46i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (3.00 - 8.26i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-5.39 - 4.53i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (10.7 - 18.6i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (22.4 - 12.9i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-40.0 - 7.05i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (6.59 - 5.53i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-11.2 - 1.98i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (-6.02 + 10.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (57.3 + 48.1i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (5.67 - 15.5i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-10.2 + 1.80i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 6.32T + 2.80e3T^{2} \) |
| 59 | \( 1 + (16.9 + 46.4i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-0.604 - 3.42i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-40.3 + 48.1i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (8.73 + 5.04i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (13.0 + 22.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (61.6 + 73.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (2.03 + 2.41i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (23.2 + 40.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (8.79 + 3.20i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.17827036707545092351191677756, −12.32483646580913953728180847372, −11.33670542318526408689581578966, −10.57077157849830837119606375241, −8.668870880289869313660280769774, −7.61857542920509544003611691117, −6.66629158462081883541501939450, −4.55223746897089443401855226403, −3.57855685474443661713561388099, −1.84302098796582347892414126277,
2.94965600027193948774163661158, 4.37143652707310606919913774676, 5.03051504072252874461861729055, 7.18324745502404125655162684773, 8.374089662193724778980643652225, 8.610068382234077895561326389494, 11.05738645351026785053455537850, 11.44040642479132785531889873332, 12.90301289426201950710287583247, 13.74577578994656530207520554674
