| L(s) = 1 | + (0.0530 + 0.00798i)2-s + (−1.90 − 0.588i)4-s + (0.162 + 2.16i)5-s + (1.17 − 2.37i)7-s + (−0.193 − 0.0929i)8-s + (−0.00869 + 0.116i)10-s + (1.58 + 4.03i)11-s + (−0.825 + 1.03i)13-s + (0.0811 − 0.116i)14-s + (3.29 + 2.24i)16-s + (2.27 + 2.10i)17-s + (1.58 + 2.75i)19-s + (0.964 − 4.22i)20-s + (0.0517 + 0.226i)22-s + (2.82 − 2.61i)23-s + ⋯ |
| L(s) = 1 | + (0.0374 + 0.00564i)2-s + (−0.954 − 0.294i)4-s + (0.0725 + 0.968i)5-s + (0.443 − 0.896i)7-s + (−0.0682 − 0.0328i)8-s + (−0.00274 + 0.0366i)10-s + (0.477 + 1.21i)11-s + (−0.229 + 0.287i)13-s + (0.0216 − 0.0310i)14-s + (0.822 + 0.560i)16-s + (0.551 + 0.511i)17-s + (0.364 + 0.631i)19-s + (0.215 − 0.945i)20-s + (0.0110 + 0.0482i)22-s + (0.588 − 0.546i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06951 + 0.515304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06951 + 0.515304i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.17 + 2.37i)T \) |
| good | 2 | \( 1 + (-0.0530 - 0.00798i)T + (1.91 + 0.589i)T^{2} \) |
| 5 | \( 1 + (-0.162 - 2.16i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.58 - 4.03i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (0.825 - 1.03i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.27 - 2.10i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.58 - 2.75i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.61i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (1.94 - 8.53i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (3.79 - 6.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.95 + 1.83i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (7.42 + 3.57i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-6.58 + 3.17i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-5.41 - 0.816i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (7.57 + 2.33i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.145 + 1.93i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 3.20i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.578 + 1.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.19 - 5.24i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (8.53 - 1.28i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (4.52 + 7.83i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.87 + 8.62i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (4.10 - 10.4i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 0.385T + 97T^{2} \) |
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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
−10.92689367955174359249980560529, −10.36946914242139315825296541997, −9.618939536681676285725819179959, −8.601883310989785809812676031150, −7.38658123428956278787038355284, −6.80618114754297273323549199291, −5.38890876407029051662763551772, −4.39677485468370978387534626187, −3.43863067007071227976560800336, −1.53196648962090235723500855947,
0.876341127817519725362131731140, 2.92190021235723866247219977739, 4.25987795855458332589326020281, 5.25718277482881304166047972658, 5.86694973102855698593905043267, 7.66900468835081006110981132637, 8.413426668932955286228549092735, 9.168589708323653801120737043574, 9.655299472332573026546685531483, 11.31674090788734223133959280872
