| 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
−11.31674090788734223133959280872, −9.655299472332573026546685531483, −9.168589708323653801120737043574, −8.413426668932955286228549092735, −7.66900468835081006110981132637, −5.86694973102855698593905043267, −5.25718277482881304166047972658, −4.25987795855458332589326020281, −2.92190021235723866247219977739, −0.876341127817519725362131731140,
1.53196648962090235723500855947, 3.43863067007071227976560800336, 4.39677485468370978387534626187, 5.38890876407029051662763551772, 6.80618114754297273323549199291, 7.38658123428956278787038355284, 8.601883310989785809812676031150, 9.618939536681676285725819179959, 10.36946914242139315825296541997, 10.92689367955174359249980560529
