| L(s) = 1 | + (−0.813 + 1.15i)2-s + (−1.73 − 0.0681i)3-s + (−0.676 − 1.88i)4-s + (1.66 + 0.889i)5-s + (1.48 − 1.94i)6-s + (−2.12 − 0.422i)7-s + (2.72 + 0.748i)8-s + (2.99 + 0.235i)9-s + (−2.38 + 1.20i)10-s + (−2.38 + 1.95i)11-s + (1.04 + 3.30i)12-s + (−1.07 + 0.575i)13-s + (2.21 − 2.11i)14-s + (−2.81 − 1.65i)15-s + (−3.08 + 2.54i)16-s + (−3.03 + 1.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.575 + 0.817i)2-s + (−0.999 − 0.0393i)3-s + (−0.338 − 0.941i)4-s + (0.743 + 0.397i)5-s + (0.606 − 0.794i)6-s + (−0.802 − 0.159i)7-s + (0.964 + 0.264i)8-s + (0.996 + 0.0786i)9-s + (−0.753 + 0.379i)10-s + (−0.719 + 0.590i)11-s + (0.300 + 0.953i)12-s + (−0.298 + 0.159i)13-s + (0.592 − 0.564i)14-s + (−0.727 − 0.426i)15-s + (−0.771 + 0.636i)16-s + (−0.736 + 0.305i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0353773 - 0.108882i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0353773 - 0.108882i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.813 - 1.15i)T \) |
| 3 | \( 1 + (1.73 + 0.0681i)T \) |
| good | 5 | \( 1 + (-1.66 - 0.889i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (2.12 + 0.422i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (2.38 - 1.95i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (1.07 - 0.575i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (3.03 - 1.25i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.0115 + 0.0381i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (4.68 + 3.12i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (2.81 + 2.31i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (0.683 - 0.683i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.24 - 4.09i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (8.68 + 5.80i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.213 - 2.16i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (3.02 - 1.25i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (5.37 - 4.41i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-10.6 - 5.68i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.0413 + 0.419i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (11.3 - 1.11i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-5.72 - 1.13i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-3.00 - 15.1i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-3.13 + 7.57i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-2.01 + 6.63i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (1.67 + 2.50i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (0.891 + 0.891i)T + 97iT^{2} \) |
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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
−11.74507384343608548759865770199, −10.46541971984826889510855861109, −10.18033633234615487346067478742, −9.368591961015243199923420561899, −7.989026955522811044705924415382, −6.88494022700100562962628668085, −6.36579146252233227201922318370, −5.46744736379297293186447013508, −4.36046930620998236781024556762, −2.06302377521304376977815376626,
0.097791012593176127913076260741, 1.90984342987331630037999549030, 3.45296702927892507545520318655, 4.90899860513442423101821010725, 5.85709876958058616606293850294, 6.98779796368361982180015497468, 8.173751780330126318596050848480, 9.441322900298145959086637417785, 9.833982446090230549403808621253, 10.80099032919652509393815552248
