| L(s) = 1 | + (−0.478 − 1.33i)2-s + (1.39 − 1.02i)3-s + (−1.54 + 1.27i)4-s + (0.700 + 1.39i)5-s + (−2.03 − 1.36i)6-s + (3.95 + 1.18i)7-s + (2.43 + 1.44i)8-s + (0.886 − 2.86i)9-s + (1.52 − 1.59i)10-s + (0.366 + 6.28i)11-s + (−0.841 + 3.36i)12-s + (−1.88 − 2.53i)13-s + (−0.316 − 5.83i)14-s + (2.40 + 1.22i)15-s + (0.756 − 3.92i)16-s + (−0.154 + 0.184i)17-s + ⋯ |
| L(s) = 1 | + (−0.338 − 0.941i)2-s + (0.804 − 0.593i)3-s + (−0.771 + 0.636i)4-s + (0.313 + 0.623i)5-s + (−0.830 − 0.556i)6-s + (1.49 + 0.448i)7-s + (0.860 + 0.510i)8-s + (0.295 − 0.955i)9-s + (0.480 − 0.505i)10-s + (0.110 + 1.89i)11-s + (−0.242 + 0.970i)12-s + (−0.523 − 0.703i)13-s + (−0.0846 − 1.55i)14-s + (0.622 + 0.316i)15-s + (0.189 − 0.981i)16-s + (−0.0374 + 0.0446i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40257 - 0.710589i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40257 - 0.710589i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.478 + 1.33i)T \) |
| 3 | \( 1 + (-1.39 + 1.02i)T \) |
| good | 5 | \( 1 + (-0.700 - 1.39i)T + (-2.98 + 4.01i)T^{2} \) |
| 7 | \( 1 + (-3.95 - 1.18i)T + (5.84 + 3.84i)T^{2} \) |
| 11 | \( 1 + (-0.366 - 6.28i)T + (-10.9 + 1.27i)T^{2} \) |
| 13 | \( 1 + (1.88 + 2.53i)T + (-3.72 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.154 - 0.184i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (0.281 + 0.335i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (0.516 + 1.72i)T + (-19.2 + 12.6i)T^{2} \) |
| 29 | \( 1 + (8.08 - 3.48i)T + (19.9 - 21.0i)T^{2} \) |
| 31 | \( 1 + (6.47 + 6.11i)T + (1.80 + 30.9i)T^{2} \) |
| 37 | \( 1 + (0.364 - 2.06i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (1.29 + 11.0i)T + (-39.8 + 9.45i)T^{2} \) |
| 43 | \( 1 + (0.947 + 1.44i)T + (-17.0 + 39.4i)T^{2} \) |
| 47 | \( 1 + (-5.93 - 6.29i)T + (-2.73 + 46.9i)T^{2} \) |
| 53 | \( 1 + (2.92 + 1.68i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.150 + 2.58i)T + (-58.6 - 6.84i)T^{2} \) |
| 61 | \( 1 + (-9.30 + 2.20i)T + (54.5 - 27.3i)T^{2} \) |
| 67 | \( 1 + (-4.95 - 2.13i)T + (45.9 + 48.7i)T^{2} \) |
| 71 | \( 1 + (7.78 - 2.83i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.51 - 0.552i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.04 - 8.92i)T + (-76.8 - 18.2i)T^{2} \) |
| 83 | \( 1 + (3.10 + 0.362i)T + (80.7 + 19.1i)T^{2} \) |
| 89 | \( 1 + (-3.76 + 10.3i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.0945 + 0.0474i)T + (57.9 + 77.8i)T^{2} \) |
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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
−11.51193857718146646978360217938, −10.51437415300982603076008991817, −9.634423457889202110664162634963, −8.763158766857525015146754331692, −7.71122282777491600045679536706, −7.19303578371926393275128797422, −5.23782028329109792177997707518, −4.04100503381883482294901959987, −2.40849242134434929389180378541, −1.85583168274753541094269342724,
1.55297654591870964292325056786, 3.80255036820280738918992577515, 4.89076806254564371096005638386, 5.63722479514294419696574099169, 7.27859271393123371559741065281, 8.152048604696779020477419249860, 8.786274583925108380410480854200, 9.472255560998351420820994946447, 10.71993999037869879050562790261, 11.37590627166775368841091597393
