| L(s) = 1 | + (1.41 + 0.0626i)2-s + (−1.72 + 0.0889i)3-s + (1.99 + 0.177i)4-s + (−1.34 − 0.719i)5-s + (−2.44 + 0.0172i)6-s + (2.49 + 0.497i)7-s + (2.80 + 0.375i)8-s + (2.98 − 0.307i)9-s + (−1.85 − 1.10i)10-s + (0.433 − 0.355i)11-s + (−3.46 − 0.129i)12-s + (0.780 − 0.417i)13-s + (3.50 + 0.859i)14-s + (2.39 + 1.12i)15-s + (3.93 + 0.705i)16-s + (4.49 − 1.86i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0443i)2-s + (−0.998 + 0.0513i)3-s + (0.996 + 0.0885i)4-s + (−0.601 − 0.321i)5-s + (−0.999 + 0.00702i)6-s + (0.944 + 0.187i)7-s + (0.991 + 0.132i)8-s + (0.994 − 0.102i)9-s + (−0.586 − 0.348i)10-s + (0.130 − 0.107i)11-s + (−0.999 − 0.0372i)12-s + (0.216 − 0.115i)13-s + (0.935 + 0.229i)14-s + (0.617 + 0.290i)15-s + (0.984 + 0.176i)16-s + (1.08 − 0.451i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96028 - 0.0132465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96028 - 0.0132465i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0626i)T \) |
| 3 | \( 1 + (1.72 - 0.0889i)T \) |
| good | 5 | \( 1 + (1.34 + 0.719i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (-2.49 - 0.497i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.433 + 0.355i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.780 + 0.417i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-4.49 + 1.86i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.781 - 2.57i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-2.97 - 1.98i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.665 - 0.546i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (0.843 - 0.843i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.46 + 4.84i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (5.55 + 3.71i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.716 - 7.27i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (10.0 - 4.17i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (5.81 - 4.77i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (1.73 + 0.926i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.965 + 9.80i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (6.69 - 0.659i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-10.3 - 2.06i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.0239 - 0.120i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (3.63 - 8.77i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.47 + 8.14i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-0.873 - 1.30i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (12.4 + 12.4i)T + 97iT^{2} \) |
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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
−11.46860270128935834245824023484, −10.92866237599958992861748974138, −9.820903122848135691685219383060, −8.183347783247703552544353264327, −7.45957010743744945612523430838, −6.28841334127404436533159698818, −5.31207601723694953017191048821, −4.64163880918211558696697771541, −3.50044316503793289091810524963, −1.48149427977909995036518204658,
1.54361101211750903794538944752, 3.45883102450189474208574884099, 4.56991353914841514519481632809, 5.32098857001394554277167715069, 6.47466909051549568955786669054, 7.29134273302548985630695099727, 8.184383052977519681308460454717, 9.999170194824299237763082982253, 10.86671410037843831842869730267, 11.47108304497667891892712592947
