| L(s) = 1 | + (0.298 − 0.222i)2-s + (1.70 − 0.283i)3-s + (−0.533 + 1.78i)4-s + (0.0404 + 0.694i)5-s + (0.447 − 0.464i)6-s + (−1.85 + 1.88i)7-s + (0.492 + 1.35i)8-s + (2.83 − 0.968i)9-s + (0.166 + 0.198i)10-s + (−3.40 + 5.18i)11-s + (−0.407 + 3.19i)12-s + (−0.147 − 0.110i)13-s + (−0.132 + 0.976i)14-s + (0.265 + 1.17i)15-s + (−2.66 − 1.75i)16-s + (−0.700 − 3.97i)17-s + ⋯ |
| L(s) = 1 | + (0.211 − 0.157i)2-s + (0.986 − 0.163i)3-s + (−0.266 + 0.891i)4-s + (0.0180 + 0.310i)5-s + (0.182 − 0.189i)6-s + (−0.699 + 0.714i)7-s + (0.173 + 0.477i)8-s + (0.946 − 0.322i)9-s + (0.0526 + 0.0627i)10-s + (−1.02 + 1.56i)11-s + (−0.117 + 0.923i)12-s + (−0.0410 − 0.0305i)13-s + (−0.0355 + 0.261i)14-s + (0.0686 + 0.303i)15-s + (−0.665 − 0.437i)16-s + (−0.169 − 0.963i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35167 + 1.18990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35167 + 1.18990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 + 0.283i)T \) |
| 7 | \( 1 + (1.85 - 1.88i)T \) |
| good | 2 | \( 1 + (-0.298 + 0.222i)T + (0.573 - 1.91i)T^{2} \) |
| 5 | \( 1 + (-0.0404 - 0.694i)T + (-4.96 + 0.580i)T^{2} \) |
| 11 | \( 1 + (3.40 - 5.18i)T + (-4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (0.147 + 0.110i)T + (3.72 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.700 + 3.97i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (1.33 + 0.235i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-0.893 - 3.76i)T + (-20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-4.61 - 1.99i)T + (19.9 + 21.0i)T^{2} \) |
| 31 | \( 1 + (0.493 + 2.08i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (-4.88 + 4.09i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-9.02 - 1.05i)T + (39.8 + 9.45i)T^{2} \) |
| 43 | \( 1 + (-2.68 - 1.76i)T + (17.0 + 39.4i)T^{2} \) |
| 47 | \( 1 + (4.32 + 1.02i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 - 9.13iT - 53T^{2} \) |
| 59 | \( 1 + (2.30 + 1.15i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (-14.7 + 4.41i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (2.57 + 0.300i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (-1.28 + 3.52i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.11 - 1.25i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-10.0 - 13.5i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (11.3 - 1.32i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (8.24 + 6.91i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.37 - 2.74i)T + (-57.9 + 77.8i)T^{2} \) |
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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
−10.95528585860446143696368440845, −9.707393287421782716306200078430, −9.309583233379403543712128409152, −8.240411092047781925146857746698, −7.44776142052167592985025658471, −6.79060792354870651854304485118, −5.12114404995672427256470509187, −4.14654234031578304265904565487, −2.82607852448183584723416338969, −2.44781171844726460995334469278,
0.877061316000263269773410281518, 2.69370494908674571342297526719, 3.84572236589028627102581014268, 4.80588585013963643280245167122, 5.98465844967668856204035141676, 6.83051847695500234705139604482, 8.143630178028249732431935257246, 8.705599540285844085109159804345, 9.710918313070681056222240479853, 10.48369882408309842599101713024
