L(s) = 1 | + (−1.37 + 1.05i)3-s + (−3.35 − 1.93i)5-s + (1.14 + 1.99i)7-s + (0.770 − 2.89i)9-s + (−3.60 + 2.08i)11-s + (0.846 + 0.488i)13-s + (6.65 − 0.884i)15-s − 6.79·17-s − 0.729i·19-s + (−3.68 − 1.52i)21-s + (2.94 − 5.10i)23-s + (5.02 + 8.69i)25-s + (2.00 + 4.79i)27-s + (4.32 − 2.49i)29-s + (1.53 − 2.65i)31-s + ⋯ |
L(s) = 1 | + (−0.792 + 0.609i)3-s + (−1.50 − 0.867i)5-s + (0.434 + 0.752i)7-s + (0.256 − 0.966i)9-s + (−1.08 + 0.627i)11-s + (0.234 + 0.135i)13-s + (1.71 − 0.228i)15-s − 1.64·17-s − 0.167i·19-s + (−0.803 − 0.331i)21-s + (0.614 − 1.06i)23-s + (1.00 + 1.73i)25-s + (0.385 + 0.922i)27-s + (0.803 − 0.463i)29-s + (0.275 − 0.476i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6690130677\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6690130677\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.37 - 1.05i)T \) |
good | 5 | \( 1 + (3.35 + 1.93i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.14 - 1.99i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.60 - 2.08i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.846 - 0.488i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 + 0.729iT - 19T^{2} \) |
| 23 | \( 1 + (-2.94 + 5.10i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.32 + 2.49i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.53 + 2.65i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.11iT - 37T^{2} \) |
| 41 | \( 1 + (3.12 - 5.41i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.80 + 2.77i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.86 - 8.42i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6.81iT - 53T^{2} \) |
| 59 | \( 1 + (-10.6 - 6.14i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 6.23i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0524 + 0.0302i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + 1.71T + 73T^{2} \) |
| 79 | \( 1 + (5.77 + 10.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.35 - 3.66i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + (4.66 + 8.08i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.810930006658532398428428755464, −8.720312566623840088315875935940, −8.431354184516700445877767661184, −7.33750788061045629457805807418, −6.39877517021253013512607785234, −5.13839935554324976838754520296, −4.68911855647198498062564586163, −4.01693581656191732565127579837, −2.50544945001392143080264201916, −0.54684569434580012620972360427,
0.70983474642353202523839182509, 2.50949121422048293626089715639, 3.69520667519345847862008138897, 4.60434113151215888781126661427, 5.57915895042497202546796728362, 6.87352986513962220310417454013, 7.13535753855306319744520362583, 8.008703205118093987059716713567, 8.590058470422453640863036362772, 10.31758156095601932680896377037