| L(s) = 1 | − 1.04·2-s − 0.901·4-s − 2.67·5-s + 1.71·7-s + 3.04·8-s + 2.80·10-s − 11-s + 3.35·13-s − 1.79·14-s − 1.38·16-s − 0.123·17-s + 1.74·19-s + 2.41·20-s + 1.04·22-s + 3.85·23-s + 2.17·25-s − 3.51·26-s − 1.54·28-s − 2.71·29-s − 8.66·31-s − 4.63·32-s + 0.129·34-s − 4.58·35-s + 5.07·37-s − 1.83·38-s − 8.14·40-s − 10.4·41-s + ⋯ |
| L(s) = 1 | − 0.741·2-s − 0.450·4-s − 1.19·5-s + 0.647·7-s + 1.07·8-s + 0.887·10-s − 0.301·11-s + 0.929·13-s − 0.479·14-s − 0.345·16-s − 0.0300·17-s + 0.401·19-s + 0.539·20-s + 0.223·22-s + 0.804·23-s + 0.434·25-s − 0.688·26-s − 0.291·28-s − 0.503·29-s − 1.55·31-s − 0.818·32-s + 0.0222·34-s − 0.775·35-s + 0.834·37-s − 0.297·38-s − 1.28·40-s − 1.63·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 13 | \( 1 - 3.35T + 13T^{2} \) |
| 17 | \( 1 + 0.123T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 - 3.85T + 23T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 + 8.66T + 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 - 1.08T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 1.85T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 1.74T + 67T^{2} \) |
| 71 | \( 1 - 3.66T + 71T^{2} \) |
| 73 | \( 1 - 1.48T + 73T^{2} \) |
| 79 | \( 1 + 6.84T + 79T^{2} \) |
| 83 | \( 1 + 3.58T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
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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
−7.51630217141208281451524856005, −7.30877783332747024658095069211, −6.16623817634527505100482921008, −5.17768543388432120482113781904, −4.70876615401665291468630292043, −3.81117044558929702101235245539, −3.36175553079184233209022595666, −1.92037545100374825681522564209, −1.02385238337914262405300984686, 0,
1.02385238337914262405300984686, 1.92037545100374825681522564209, 3.36175553079184233209022595666, 3.81117044558929702101235245539, 4.70876615401665291468630292043, 5.17768543388432120482113781904, 6.16623817634527505100482921008, 7.30877783332747024658095069211, 7.51630217141208281451524856005
