| L(s) = 1 | + (1.26 + 0.626i)2-s + (0.841 − 0.540i)3-s + (1.21 + 1.58i)4-s + (−0.329 − 0.720i)5-s + (1.40 − 0.158i)6-s + (1.17 − 0.345i)7-s + (0.544 + 2.77i)8-s + (0.415 − 0.909i)9-s + (0.0343 − 1.12i)10-s + (0.818 − 0.709i)11-s + (1.88 + 0.680i)12-s + (−0.202 + 0.688i)13-s + (1.70 + 0.299i)14-s + (−0.666 − 0.428i)15-s + (−1.04 + 3.85i)16-s + (3.72 + 0.534i)17-s + ⋯ |
| L(s) = 1 | + (0.896 + 0.443i)2-s + (0.485 − 0.312i)3-s + (0.607 + 0.794i)4-s + (−0.147 − 0.322i)5-s + (0.573 − 0.0645i)6-s + (0.444 − 0.130i)7-s + (0.192 + 0.981i)8-s + (0.138 − 0.303i)9-s + (0.0108 − 0.354i)10-s + (0.246 − 0.213i)11-s + (0.542 + 0.196i)12-s + (−0.0561 + 0.191i)13-s + (0.456 + 0.0800i)14-s + (−0.172 − 0.110i)15-s + (−0.262 + 0.964i)16-s + (0.902 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.86448 + 0.595213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.86448 + 0.595213i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 - 0.626i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (4.14 - 2.41i)T \) |
| good | 5 | \( 1 + (0.329 + 0.720i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 0.345i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.818 + 0.709i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.202 - 0.688i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-3.72 - 0.534i)T + (16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-2.00 + 0.288i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (9.25 + 1.33i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.114 + 0.178i)T + (-12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.01 - 2.21i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.46 - 3.20i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (3.13 + 4.88i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 4.85iT - 47T^{2} \) |
| 53 | \( 1 + (1.75 - 0.514i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (8.52 + 2.50i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-5.44 - 3.49i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.305 + 0.264i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (6.97 + 6.04i)T + (10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.454 - 3.16i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.57 - 2.22i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (15.0 + 6.86i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-3.02 - 4.70i)T + (-36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (1.75 - 0.801i)T + (63.5 - 73.3i)T^{2} \) |
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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
−11.18354720552099199705592711184, −9.914661805298661135591577571775, −8.781967752066992994187378330477, −7.940515696800955540708321304395, −7.32938282529730510793760132660, −6.19788627349234175333891916382, −5.25750952789200785299675624351, −4.15523589270773704043983807065, −3.22032375718447108294529625158, −1.75487486854041273239511668339,
1.69458566785492034415910922794, 3.00870085768554413741767205607, 3.88001192439478631363616071369, 4.97625549770270909086300409605, 5.85860601258623519417352969372, 7.11460524330180477804192826723, 7.907692474817845528602867738504, 9.227115141117985972435994580498, 9.983244450779999363480865727636, 10.87284659064548693544723344879
