L(s) = 1 | + (−1.71 − 0.243i)3-s + (1.34 − 2.32i)5-s + (2.48 + 4.30i)7-s + (2.88 + 0.835i)9-s + (1.26 + 2.19i)11-s + (−2.21 + 3.83i)13-s + (−2.86 + 3.65i)15-s − 2.43·17-s − 4.18·19-s + (−3.21 − 7.99i)21-s + (−0.570 + 0.988i)23-s + (−1.09 − 1.89i)25-s + (−4.73 − 2.13i)27-s + (−3.00 − 5.20i)29-s + (−2.65 + 4.59i)31-s + ⋯ |
L(s) = 1 | + (−0.990 − 0.140i)3-s + (0.599 − 1.03i)5-s + (0.939 + 1.62i)7-s + (0.960 + 0.278i)9-s + (0.382 + 0.662i)11-s + (−0.614 + 1.06i)13-s + (−0.739 + 0.943i)15-s − 0.590·17-s − 0.959·19-s + (−0.701 − 1.74i)21-s + (−0.118 + 0.206i)23-s + (−0.218 − 0.378i)25-s + (−0.911 − 0.410i)27-s + (−0.557 − 0.966i)29-s + (−0.476 + 0.825i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.115398981\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.115398981\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.71 + 0.243i)T \) |
good | 5 | \( 1 + (-1.34 + 2.32i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.48 - 4.30i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.26 - 2.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.21 - 3.83i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 2.43T + 17T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 + (0.570 - 0.988i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.00 + 5.20i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.65 - 4.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.241T + 37T^{2} \) |
| 41 | \( 1 + (3.21 - 5.56i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.57 - 9.65i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.37 + 4.11i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 + (5.40 - 9.36i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.16 + 7.20i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.13 + 1.96i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 + (3.14 + 5.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.738 - 1.27i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + (-5.89 - 10.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.795568318595734819538094272075, −9.172372417647407456067200214356, −8.576933728231110392761566909407, −7.46740879408615915528700307821, −6.36828498966722145657381627712, −5.73297064504247759099390504650, −4.78358605283028732648481637848, −4.51483588528214561901524401513, −2.14175478551536010975608728013, −1.64028821710723400303297069789,
0.54534482634922645876223930342, 1.99523860590009149600438222934, 3.56264872323583305824976308359, 4.41650125660797468132690259307, 5.40854088538519895274971801284, 6.28470837465145500997559584025, 7.09577985271203617534052626254, 7.59553497713424141513568118679, 8.841167683221484232483628269232, 10.11861261952680553374315498066