| L(s) = 1 | + (1.07 + 0.921i)2-s + (1.37 + 1.05i)3-s + (0.300 + 1.97i)4-s + (0.499 + 2.39i)6-s + (−1.5 + 2.39i)8-s + (0.771 + 2.89i)9-s + (−1.67 + 3.03i)12-s − 1.57i·13-s + (−3.81 + 1.18i)16-s + (−1.84 + 3.82i)18-s − 4.79i·23-s + (−4.59 + 1.70i)24-s + 5·25-s + (1.45 − 1.69i)26-s + (−1.99 + 4.79i)27-s + ⋯ |
| L(s) = 1 | + (0.758 + 0.651i)2-s + (0.792 + 0.609i)3-s + (0.150 + 0.988i)4-s + (0.204 + 0.978i)6-s + (−0.530 + 0.847i)8-s + (0.257 + 0.966i)9-s + (−0.483 + 0.875i)12-s − 0.437i·13-s + (−0.954 + 0.297i)16-s + (−0.434 + 0.900i)18-s − 0.999i·23-s + (−0.937 + 0.349i)24-s + 25-s + (0.284 − 0.331i)26-s + (−0.384 + 0.922i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55188 + 2.23397i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55188 + 2.23397i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.07 - 0.921i)T \) |
| 3 | \( 1 + (-1.37 - 1.05i)T \) |
| 23 | \( 1 + 4.79iT \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 1.57iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 2.64iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 13.7iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 1.04iT - 71T^{2} \) |
| 73 | \( 1 - 9.44T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.00396896726113300343716040165, −10.17342653825763510705244585632, −9.012359753595340004543405404682, −8.356458527134145492188920428921, −7.50491025206185193128276781379, −6.50277830151894921491787823982, −5.29777537608637311753708079234, −4.47611547014437323758337970922, −3.44491113404358681511823150114, −2.45190927733745579228604031249,
1.30898193353791838536453978326, 2.54522710359210399171294487353, 3.54130939077545999881427635171, 4.61106166882319249413920561198, 5.89919609384879735731607621384, 6.80747315774303119660082727899, 7.74542976261657058884756953395, 8.984175562179244531871305676643, 9.576233223440541077379311335758, 10.66092854006333508458422410305
