L(s) = 1 | + (1.71 + 0.231i)3-s + (1.74 + 3.01i)5-s + (−1.80 − 1.04i)7-s + (2.89 + 0.795i)9-s + (0.116 + 0.0675i)11-s + (−2.63 + 1.52i)13-s + (2.29 + 5.58i)15-s − 4.19i·17-s − 0.919·19-s + (−2.86 − 2.21i)21-s + (−0.689 − 1.19i)23-s + (−3.57 + 6.19i)25-s + (4.78 + 2.03i)27-s + (4.24 − 7.34i)29-s + (4.39 − 2.53i)31-s + ⋯ |
L(s) = 1 | + (0.990 + 0.133i)3-s + (0.779 + 1.35i)5-s + (−0.683 − 0.394i)7-s + (0.964 + 0.265i)9-s + (0.0352 + 0.0203i)11-s + (−0.731 + 0.422i)13-s + (0.591 + 1.44i)15-s − 1.01i·17-s − 0.210·19-s + (−0.624 − 0.482i)21-s + (−0.143 − 0.249i)23-s + (−0.715 + 1.23i)25-s + (0.919 + 0.392i)27-s + (0.787 − 1.36i)29-s + (0.790 − 0.456i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71458 + 0.609153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71458 + 0.609153i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 - 0.231i)T \) |
good | 5 | \( 1 + (-1.74 - 3.01i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.80 + 1.04i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.116 - 0.0675i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.63 - 1.52i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.19iT - 17T^{2} \) |
| 19 | \( 1 + 0.919T + 19T^{2} \) |
| 23 | \( 1 + (0.689 + 1.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.24 + 7.34i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.39 + 2.53i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.61iT - 37T^{2} \) |
| 41 | \( 1 + (-1.79 + 1.03i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.41 - 9.37i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.205 + 0.356i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.968T + 53T^{2} \) |
| 59 | \( 1 + (3.88 - 2.24i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.44 + 4.29i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.15 + 5.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 4.06T + 73T^{2} \) |
| 79 | \( 1 + (10.8 + 6.27i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.23 + 3.02i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 8.35iT - 89T^{2} \) |
| 97 | \( 1 + (0.477 - 0.826i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.88891503973310737687121598806, −10.66762664593687614011412845859, −9.828707263784644535013379935036, −9.472776834679652722247508408844, −7.955004865283578807650158434201, −6.98964315635443342716626588603, −6.30551824228057273128382611866, −4.52130369979231935234737442710, −3.12305752427886770923808050774, −2.34123238846219232447366373510,
1.57318145223686444026515278702, 2.98910087280652977677542355886, 4.49024961414475425071345910128, 5.62770801384245256260189738997, 6.82596042192178638041157570983, 8.226878250461092380227217058747, 8.817232832205518058058147255828, 9.656846977477518635900959036038, 10.35663969540838784997699661454, 12.30348197641022016537430294482