| L(s) = 1 | + (1.93 − 0.503i)2-s + (3.49 − 1.94i)4-s + (−3.67 + 1.33i)5-s + (2.42 − 0.428i)7-s + (5.78 − 5.52i)8-s + (−6.43 + 4.43i)10-s + (6.11 − 16.8i)11-s + (15.7 + 13.2i)13-s + (4.48 − 2.05i)14-s + (8.41 − 13.6i)16-s + (11.4 − 19.8i)17-s + (6.65 − 3.83i)19-s + (−10.2 + 11.8i)20-s + (3.38 − 35.6i)22-s + (−1.94 − 0.343i)23-s + ⋯ |
| L(s) = 1 | + (0.967 − 0.251i)2-s + (0.873 − 0.486i)4-s + (−0.734 + 0.267i)5-s + (0.346 − 0.0611i)7-s + (0.722 − 0.690i)8-s + (−0.643 + 0.443i)10-s + (0.556 − 1.52i)11-s + (1.21 + 1.01i)13-s + (0.320 − 0.146i)14-s + (0.525 − 0.850i)16-s + (0.674 − 1.16i)17-s + (0.350 − 0.202i)19-s + (−0.511 + 0.590i)20-s + (0.153 − 1.61i)22-s + (−0.0846 − 0.0149i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.74373 - 1.12032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74373 - 1.12032i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.93 + 0.503i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.67 - 1.33i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (-2.42 + 0.428i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-6.11 + 16.8i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-15.7 - 13.2i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-11.4 + 19.8i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.65 + 3.83i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (1.94 + 0.343i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (33.6 - 28.2i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-20.3 - 3.58i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (15.3 - 26.6i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-4.57 - 3.83i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-10.9 + 29.9i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (53.8 - 9.49i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 19.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-19.2 - 52.7i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (2.78 + 15.7i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.68i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-86.0 - 49.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-31.8 - 55.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (12.9 + 15.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (33.4 + 39.9i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-8.30 - 14.3i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (120. + 43.6i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.41473597199892602156922921776, −10.93611327809718485216786603760, −9.482572336109990747112274585500, −8.350799341242134089893176028628, −7.20958017061865195501132775049, −6.27872240075301678591759993140, −5.20114843168921055758112098910, −3.87910093329544112176585939273, −3.19517132146790117988078281454, −1.25158874964792794091680859977,
1.72517598826565302401895834090, 3.55310952706893060499071532470, 4.29336756015728994904771998371, 5.49397305669168949458057288087, 6.49865815058342289538813172524, 7.80003992052683554957586493086, 8.157835533232052135400741709935, 9.778677970590549962547184596308, 10.88461565684159089003301487741, 11.74254740734870222199631695741
