| L(s) = 1 | + (−0.787 + 1.06i)2-s + (2.66 − 0.505i)3-s + (0.0709 + 0.230i)4-s + (−0.248 − 2.22i)5-s + (−1.56 + 3.24i)6-s + (1.25 + 2.33i)7-s + (−2.80 − 0.981i)8-s + (4.07 − 1.60i)9-s + (2.56 + 1.48i)10-s + (−0.869 + 2.21i)11-s + (0.305 + 0.578i)12-s + (4.06 + 0.458i)13-s + (−3.47 − 0.499i)14-s + (−1.78 − 5.80i)15-s + (2.85 − 1.94i)16-s + (−2.60 − 0.0973i)17-s + ⋯ |
| L(s) = 1 | + (−0.556 + 0.754i)2-s + (1.54 − 0.291i)3-s + (0.0354 + 0.115i)4-s + (−0.111 − 0.993i)5-s + (−0.638 + 1.32i)6-s + (0.473 + 0.880i)7-s + (−0.991 − 0.347i)8-s + (1.35 − 0.533i)9-s + (0.811 + 0.469i)10-s + (−0.262 + 0.668i)11-s + (0.0882 + 0.166i)12-s + (1.12 + 0.127i)13-s + (−0.928 − 0.133i)14-s + (−0.460 − 1.49i)15-s + (0.714 − 0.487i)16-s + (−0.631 − 0.0236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40449 + 0.611793i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40449 + 0.611793i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.248 + 2.22i)T \) |
| 7 | \( 1 + (-1.25 - 2.33i)T \) |
| good | 2 | \( 1 + (0.787 - 1.06i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-2.66 + 0.505i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (0.869 - 2.21i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-4.06 - 0.458i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (2.60 + 0.0973i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-2.06 + 3.56i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.136 + 3.64i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-0.312 + 0.0712i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (5.82 - 3.36i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.01 - 2.64i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-4.47 - 9.29i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (3.44 + 9.83i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (4.50 + 3.32i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (10.3 + 5.47i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.105 - 1.41i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-4.27 + 13.8i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-1.24 - 0.333i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.45 + 6.39i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (4.61 - 3.40i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-5.92 - 3.42i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.720 + 6.39i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-5.24 - 13.3i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (1.25 + 1.25i)T + 97iT^{2} \) |
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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
−12.54328403816339422202877084598, −11.41821078992444718046047941913, −9.572864215760888042197750060497, −8.825658377769931276667183258071, −8.476537739737307314584598913502, −7.67878326979420931172256305350, −6.55117785637538873758314809317, −4.96633992313615399978766877283, −3.42825600253078565857867523483, −1.99482588719819610245818258662,
1.75225898956468800056242426451, 3.11786955412149017196286809330, 3.83506821308242692298984585313, 5.95064959854273345646117743185, 7.42133696678571816044582368035, 8.255044283079111452056407558535, 9.176980902609793154020451103985, 10.10658926426892664236102660948, 10.85703044355739046009171196101, 11.42527928732648930656614484093
