L(s) = 1 | + 1.18·2-s + 2.42·3-s − 0.605·4-s + 5-s + 2.86·6-s − 4.73·7-s − 3.07·8-s + 2.89·9-s + 1.18·10-s + 0.245·11-s − 1.46·12-s − 2.62·13-s − 5.59·14-s + 2.42·15-s − 2.42·16-s − 4.98·17-s + 3.42·18-s − 6.53·19-s − 0.605·20-s − 11.5·21-s + 0.289·22-s − 1.75·23-s − 7.47·24-s + 25-s − 3.09·26-s − 0.250·27-s + 2.86·28-s + ⋯ |
L(s) = 1 | + 0.835·2-s + 1.40·3-s − 0.302·4-s + 0.447·5-s + 1.17·6-s − 1.78·7-s − 1.08·8-s + 0.965·9-s + 0.373·10-s + 0.0739·11-s − 0.424·12-s − 0.727·13-s − 1.49·14-s + 0.627·15-s − 0.605·16-s − 1.21·17-s + 0.806·18-s − 1.50·19-s − 0.135·20-s − 2.50·21-s + 0.0617·22-s − 0.365·23-s − 1.52·24-s + 0.200·25-s − 0.607·26-s − 0.0481·27-s + 0.541·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 3 | \( 1 - 2.42T + 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 0.245T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 + 6.53T + 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 + 6.93T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 0.935T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 2.96T + 59T^{2} \) |
| 61 | \( 1 - 4.11T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 + 6.29T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−9.046290664521083713562615429006, −8.173248992970089746686877981729, −7.00185011246321542144416534487, −6.37529524107281539434936374316, −5.59413456505015638016433452662, −4.24366236611421923115168716225, −3.87253109026382148126872296582, −2.71828108980449821292014911088, −2.42095218145822162898967627764, 0,
2.42095218145822162898967627764, 2.71828108980449821292014911088, 3.87253109026382148126872296582, 4.24366236611421923115168716225, 5.59413456505015638016433452662, 6.37529524107281539434936374316, 7.00185011246321542144416534487, 8.173248992970089746686877981729, 9.046290664521083713562615429006