| L(s) = 1 | − 2-s + 1.47·3-s + 4-s − 3.13·5-s − 1.47·6-s + 0.874·7-s − 8-s − 0.829·9-s + 3.13·10-s − 1.07·11-s + 1.47·12-s + 0.208·13-s − 0.874·14-s − 4.61·15-s + 16-s + 6.17·17-s + 0.829·18-s − 0.651·19-s − 3.13·20-s + 1.28·21-s + 1.07·22-s + 3.64·23-s − 1.47·24-s + 4.80·25-s − 0.208·26-s − 5.64·27-s + 0.874·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.850·3-s + 0.5·4-s − 1.40·5-s − 0.601·6-s + 0.330·7-s − 0.353·8-s − 0.276·9-s + 0.990·10-s − 0.322·11-s + 0.425·12-s + 0.0577·13-s − 0.233·14-s − 1.19·15-s + 0.250·16-s + 1.49·17-s + 0.195·18-s − 0.149·19-s − 0.700·20-s + 0.281·21-s + 0.228·22-s + 0.760·23-s − 0.300·24-s + 0.961·25-s − 0.0408·26-s − 1.08·27-s + 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 5 | \( 1 + 3.13T + 5T^{2} \) |
| 7 | \( 1 - 0.874T + 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 0.208T + 13T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 + 0.651T + 19T^{2} \) |
| 23 | \( 1 - 3.64T + 23T^{2} \) |
| 29 | \( 1 + 1.56T + 29T^{2} \) |
| 37 | \( 1 + 2.49T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 - 1.05T + 53T^{2} \) |
| 59 | \( 1 - 1.86T + 59T^{2} \) |
| 61 | \( 1 - 4.60T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 0.327T + 73T^{2} \) |
| 79 | \( 1 + 1.90T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 0.881T + 89T^{2} \) |
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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
−7.944083179398879074119572294775, −7.42475237299942830565084294794, −6.61437345025545014439345846611, −5.54359794003886928080819844953, −4.79249706312436738940827339008, −3.61982148015395216642470287886, −3.32984615781250748976046434515, −2.38026355986842188130019845107, −1.21032370997823864188493828699, 0,
1.21032370997823864188493828699, 2.38026355986842188130019845107, 3.32984615781250748976046434515, 3.61982148015395216642470287886, 4.79249706312436738940827339008, 5.54359794003886928080819844953, 6.61437345025545014439345846611, 7.42475237299942830565084294794, 7.944083179398879074119572294775
