| L(s) = 1 | − 2.11·3-s − 2.66·5-s + 1.46·7-s + 1.45·9-s − 1.58i·11-s − 3.11i·13-s + 5.62·15-s − 1.72i·17-s + 3.35i·19-s − 3.09·21-s + (4.35 − 2.01i)23-s + 2.09·25-s + 3.25·27-s − 5.01i·29-s + 3.47i·31-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 1.19·5-s + 0.554·7-s + 0.486·9-s − 0.477i·11-s − 0.863i·13-s + 1.45·15-s − 0.419i·17-s + 0.770i·19-s − 0.675·21-s + (0.907 − 0.420i)23-s + 0.419·25-s + 0.626·27-s − 0.931i·29-s + 0.624i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1764277807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1764277807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-4.35 + 2.01i)T \) |
| good | 3 | \( 1 + 2.11T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 + 1.58iT - 11T^{2} \) |
| 13 | \( 1 + 3.11iT - 13T^{2} \) |
| 17 | \( 1 + 1.72iT - 17T^{2} \) |
| 19 | \( 1 - 3.35iT - 19T^{2} \) |
| 29 | \( 1 + 5.01iT - 29T^{2} \) |
| 31 | \( 1 - 3.47iT - 31T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 - 2.59iT - 43T^{2} \) |
| 47 | \( 1 + 0.722iT - 47T^{2} \) |
| 53 | \( 1 - 5.13T + 53T^{2} \) |
| 59 | \( 1 + 0.758T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 67 | \( 1 - 16.0iT - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 7.61T + 79T^{2} \) |
| 83 | \( 1 - 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 7.61iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−10.11324647742892312259347259640, −8.774270062001800367039997024606, −8.125930835368384565101799920451, −7.39536553644078957389484877229, −6.49167409118238115250199863492, −5.55363278575305244086876031858, −4.93598183945525487917473301924, −3.97721159409043735907219107251, −2.93476364467590711974337894663, −1.05749842186402631805330335025,
0.10330441747801744119435737341, 1.60995270070396635100983836668, 3.26492793504023566622580435667, 4.42366965562196250080085320575, 4.88116082613720623392001341536, 5.87303218209380850005738592557, 6.93085061137695694546340376891, 7.34111097365946185567297327137, 8.418732927089241981102217844421, 9.111734372120372362467481679390
