| L(s) = 1 | + (−1.00 − 0.801i)2-s + (−1.06 − 1.36i)3-s + (−0.0769 − 0.337i)4-s + (0.271 + 3.62i)5-s + (−0.0228 + 2.22i)6-s + (1.01 − 2.44i)7-s + (−1.30 + 2.71i)8-s + (−0.727 + 2.91i)9-s + (2.63 − 3.86i)10-s + (3.96 − 1.55i)11-s + (−0.378 + 0.464i)12-s + (−0.639 + 0.251i)13-s + (−2.98 + 1.64i)14-s + (4.66 − 4.23i)15-s + (2.87 − 1.38i)16-s + (2.80 − 0.864i)17-s + ⋯ |
| L(s) = 1 | + (−0.711 − 0.567i)2-s + (−0.615 − 0.788i)3-s + (−0.0384 − 0.168i)4-s + (0.121 + 1.62i)5-s + (−0.00934 + 0.909i)6-s + (0.384 − 0.923i)7-s + (−0.462 + 0.961i)8-s + (−0.242 + 0.970i)9-s + (0.833 − 1.22i)10-s + (1.19 − 0.468i)11-s + (−0.109 + 0.134i)12-s + (−0.177 + 0.0696i)13-s + (−0.796 + 0.438i)14-s + (1.20 − 1.09i)15-s + (0.718 − 0.345i)16-s + (0.679 − 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.664944 - 0.498445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.664944 - 0.498445i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.06 + 1.36i)T \) |
| 7 | \( 1 + (-1.01 + 2.44i)T \) |
| good | 2 | \( 1 + (1.00 + 0.801i)T + (0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.271 - 3.62i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-3.96 + 1.55i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (0.639 - 0.251i)T + (9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-2.80 + 0.864i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-5.82 - 3.36i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.05 + 3.29i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.899 - 2.91i)T + (-23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + 7.99iT - 31T^{2} \) |
| 37 | \( 1 + (5.70 + 5.29i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-7.09 + 4.84i)T + (14.9 - 38.1i)T^{2} \) |
| 43 | \( 1 + (-1.74 - 1.18i)T + (15.7 + 40.0i)T^{2} \) |
| 47 | \( 1 + (1.23 - 1.55i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-2.49 - 2.68i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (3.70 - 1.78i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-12.0 - 2.75i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + 0.527T + 67T^{2} \) |
| 71 | \( 1 + (-5.75 + 1.31i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-12.7 - 5.00i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 + (0.245 - 0.625i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-9.03 + 1.36i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-3.13 + 1.81i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.90701706043596675118421837143, −10.30311333736642781327631988943, −9.472373031728282947955305714521, −8.000010078183826743638069822145, −7.28038529578931293198464944257, −6.37612708423253928915329590277, −5.52483777322448151750708454320, −3.70133938519619107248221990709, −2.27921658501495107361334571122, −0.973203636586494016442169782324,
1.12699867197841102675438205119, 3.60637710762096636485944794374, 4.79075743345148577630307821273, 5.48794285414994102336423905895, 6.61178538779913683487910292136, 7.947897729370404935257148759633, 8.784128880554840612733051979627, 9.393086822843251957364199188550, 9.827522861890285485659746700293, 11.55963613005289141855470802594
