L(s) = 1 | − 1.86·2-s − 3-s + 1.46·4-s − 3.32·5-s + 1.86·6-s + 1.13·7-s + 8-s + 9-s + 6.18·10-s + 0.398·11-s − 1.46·12-s + 2.39·13-s − 2.11·14-s + 3.32·15-s − 4.78·16-s + 7.10·17-s − 1.86·18-s + 1.53·19-s − 4.86·20-s − 1.13·21-s − 0.740·22-s + 3.25·23-s − 24-s + 6.04·25-s − 4.46·26-s − 27-s + 1.66·28-s + ⋯ |
L(s) = 1 | − 1.31·2-s − 0.577·3-s + 0.731·4-s − 1.48·5-s + 0.759·6-s + 0.430·7-s + 0.353·8-s + 0.333·9-s + 1.95·10-s + 0.120·11-s − 0.422·12-s + 0.665·13-s − 0.566·14-s + 0.858·15-s − 1.19·16-s + 1.72·17-s − 0.438·18-s + 0.352·19-s − 1.08·20-s − 0.248·21-s − 0.157·22-s + 0.679·23-s − 0.204·24-s + 1.20·25-s − 0.875·26-s − 0.192·27-s + 0.314·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4156595813\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4156595813\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 - 1.13T + 7T^{2} \) |
| 11 | \( 1 - 0.398T + 11T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 - 1.25T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 + 9.69T + 43T^{2} \) |
| 47 | \( 1 - 8.71T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 61 | \( 1 - 0.989T + 61T^{2} \) |
| 67 | \( 1 - 5.45T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 4.73T + 73T^{2} \) |
| 79 | \( 1 - 7.04T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−12.15115091323008454373695491675, −11.52832314049207541659394202048, −10.71237798456219590306607710559, −9.721217379621078642055548092576, −8.401406427558736789534259397578, −7.85009310363940692148093872438, −6.87134282413878860829818389583, −5.10041782871437357986933036727, −3.70709883862564397941779945184, −0.983967072382549312772776173028,
0.983967072382549312772776173028, 3.70709883862564397941779945184, 5.10041782871437357986933036727, 6.87134282413878860829818389583, 7.85009310363940692148093872438, 8.401406427558736789534259397578, 9.721217379621078642055548092576, 10.71237798456219590306607710559, 11.52832314049207541659394202048, 12.15115091323008454373695491675