| L(s) = 1 | + (1.29 + 0.562i)2-s + (−1.16 + 1.28i)3-s + (1.36 + 1.45i)4-s + (2.26 − 0.198i)5-s + (−2.23 + 1.00i)6-s + (2.00 + 0.728i)7-s + (0.952 + 2.66i)8-s + (−0.278 − 2.98i)9-s + (3.05 + 1.01i)10-s + (1.63 + 0.143i)11-s + (−3.46 + 0.0472i)12-s + (−1.75 − 2.50i)13-s + (2.18 + 2.07i)14-s + (−2.39 + 3.13i)15-s + (−0.262 + 3.99i)16-s + (−4.28 − 2.47i)17-s + ⋯ |
| L(s) = 1 | + (0.917 + 0.397i)2-s + (−0.673 + 0.739i)3-s + (0.683 + 0.729i)4-s + (1.01 − 0.0887i)5-s + (−0.911 + 0.410i)6-s + (0.756 + 0.275i)7-s + (0.336 + 0.941i)8-s + (−0.0927 − 0.995i)9-s + (0.965 + 0.321i)10-s + (0.493 + 0.0432i)11-s + (−0.999 + 0.0136i)12-s + (−0.487 − 0.696i)13-s + (0.584 + 0.553i)14-s + (−0.617 + 0.809i)15-s + (−0.0655 + 0.997i)16-s + (−1.03 − 0.599i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79890 + 1.51097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79890 + 1.51097i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.29 - 0.562i)T \) |
| 3 | \( 1 + (1.16 - 1.28i)T \) |
| good | 5 | \( 1 + (-2.26 + 0.198i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-2.00 - 0.728i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 0.143i)T + (10.8 + 1.91i)T^{2} \) |
| 13 | \( 1 + (1.75 + 2.50i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (4.28 + 2.47i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0566 - 0.211i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.10 - 3.03i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (3.06 - 4.38i)T + (-9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (0.867 + 2.38i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.604 + 0.161i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.89 + 10.7i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.249 + 2.85i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (8.10 + 2.95i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.90 - 1.90i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.765 - 8.74i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-12.4 + 5.80i)T + (39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (-3.08 + 2.15i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (11.5 + 6.66i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-9.80 + 5.66i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.60 + 1.16i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.27 - 5.09i)T + (28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (-4.19 - 7.26i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.12 + 1.78i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.40418763444196791128219361022, −10.70499983154701870367653177520, −9.584827973625089933776358944554, −8.736538352977829077145906335307, −7.34957415947459361668818241231, −6.33305526727950602783003701530, −5.39253009796653331235542633730, −4.93356533617165118821674032542, −3.65474528446882377700227410099, −2.10842122781352939599839942855,
1.51400735397910070959573807153, 2.35657206926702301708541339311, 4.31441515450884606208713603484, 5.14088767960005390152814697079, 6.28457491461964137597759406567, 6.69429324624779547423752932665, 7.967121928230957461864119253269, 9.427873098981177986185238660625, 10.37992303295992725574808022834, 11.29474195071218586641049495658
