L(s) = 1 | − 1.73·3-s + 5-s + 2.72·7-s + 0.0129·9-s − 2.68·11-s + 3.26·13-s − 1.73·15-s − 1.12·17-s − 5.16·19-s − 4.72·21-s + 4.43·23-s + 25-s + 5.18·27-s − 4.37·29-s − 4.77·31-s + 4.65·33-s + 2.72·35-s + 5.04·37-s − 5.67·39-s − 5.99·41-s + 10.1·43-s + 0.0129·45-s + 1.88·47-s + 0.404·49-s + 1.95·51-s − 4.64·53-s − 2.68·55-s + ⋯ |
L(s) = 1 | − 1.00·3-s + 0.447·5-s + 1.02·7-s + 0.00431·9-s − 0.809·11-s + 0.906·13-s − 0.448·15-s − 0.272·17-s − 1.18·19-s − 1.03·21-s + 0.924·23-s + 0.200·25-s + 0.997·27-s − 0.811·29-s − 0.858·31-s + 0.811·33-s + 0.459·35-s + 0.828·37-s − 0.908·39-s − 0.936·41-s + 1.54·43-s + 0.00193·45-s + 0.274·47-s + 0.0578·49-s + 0.273·51-s − 0.638·53-s − 0.361·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 4.77T + 31T^{2} \) |
| 37 | \( 1 - 5.04T + 37T^{2} \) |
| 41 | \( 1 + 5.99T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.88T + 47T^{2} \) |
| 53 | \( 1 + 4.64T + 53T^{2} \) |
| 59 | \( 1 - 2.26T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 - 9.75T + 73T^{2} \) |
| 79 | \( 1 - 7.93T + 79T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 3.63T + 97T^{2} \) |
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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
−7.48929938858144174441329040319, −6.61168064010620274944418282450, −5.98635757397480948879044118416, −5.45874403206253279970187395419, −4.82907813665419519639143674418, −4.16753073054282170732221527701, −3.01472173783386502459443243916, −2.08170005760977306816754396542, −1.21566507171091955231492897042, 0,
1.21566507171091955231492897042, 2.08170005760977306816754396542, 3.01472173783386502459443243916, 4.16753073054282170732221527701, 4.82907813665419519639143674418, 5.45874403206253279970187395419, 5.98635757397480948879044118416, 6.61168064010620274944418282450, 7.48929938858144174441329040319