L(s) = 1 | + (−0.258 − 0.965i)2-s + (0.866 − 0.5i)3-s + (−0.866 + 0.499i)4-s + (−1.56 + 0.419i)5-s + (−0.707 − 0.707i)6-s + (−2.13 + 1.56i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (0.810 + 1.40i)10-s + (−0.354 + 1.32i)11-s + (−0.5 + 0.866i)12-s + (−2.72 + 2.36i)13-s + (2.06 + 1.65i)14-s + (−1.14 + 1.14i)15-s + (0.500 − 0.866i)16-s + (1.90 + 3.29i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (0.499 − 0.288i)3-s + (−0.433 + 0.249i)4-s + (−0.700 + 0.187i)5-s + (−0.288 − 0.288i)6-s + (−0.805 + 0.591i)7-s + (0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.256 + 0.444i)10-s + (−0.106 + 0.398i)11-s + (−0.144 + 0.249i)12-s + (−0.754 + 0.656i)13-s + (0.551 + 0.442i)14-s + (−0.296 + 0.296i)15-s + (0.125 − 0.216i)16-s + (0.460 + 0.798i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.557771 + 0.404483i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.557771 + 0.404483i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (2.13 - 1.56i)T \) |
| 13 | \( 1 + (2.72 - 2.36i)T \) |
good | 5 | \( 1 + (1.56 - 0.419i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (0.354 - 1.32i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.90 - 3.29i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.67 - 0.448i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.60 - 2.08i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.0101T + 29T^{2} \) |
| 31 | \( 1 + (2.23 - 8.33i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (5.33 - 1.42i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.53 - 1.53i)T + 41iT^{2} \) |
| 43 | \( 1 + 7.52iT - 43T^{2} \) |
| 47 | \( 1 + (0.746 + 2.78i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.77 - 3.07i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.53 + 1.48i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (1.00 + 0.579i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.07 + 0.555i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-7.80 + 7.80i)T - 71iT^{2} \) |
| 73 | \( 1 + (1.16 + 0.311i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (7.13 - 12.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.61 - 3.61i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.84 - 6.89i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (5.73 + 5.73i)T + 97iT^{2} \) |
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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
−10.95569662884430243970570644335, −10.04644644446725190213886618363, −9.237700226979371476794804647137, −8.502254697716235080845040408125, −7.47078198854290268338285222841, −6.68771734372632746260924274917, −5.25253239075741111678118458988, −3.91290002421427834847218888030, −3.08107660938452444029986117829, −1.86256941603511528052581638443,
0.38909891703415361573438435828, 2.87712995670840693169551507630, 3.94646460382520233598374597143, 4.94004420163598208641769065731, 6.13203008532524311061904158860, 7.32117792934749031876597329916, 7.77318666550089233120921963395, 8.805652953008920834413059901907, 9.647811656077844517377442746235, 10.34724648404558469474826143090