L(s) = 1 | − 2-s − 3·3-s − 4-s − 2·5-s + 3·6-s − 5·7-s + 3·8-s + 6·9-s + 2·10-s − 4·11-s + 3·12-s − 7·13-s + 5·14-s + 6·15-s − 16-s − 2·17-s − 6·18-s − 5·19-s + 2·20-s + 15·21-s + 4·22-s − 2·23-s − 9·24-s − 25-s + 7·26-s − 9·27-s + 5·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s − 0.894·5-s + 1.22·6-s − 1.88·7-s + 1.06·8-s + 2·9-s + 0.632·10-s − 1.20·11-s + 0.866·12-s − 1.94·13-s + 1.33·14-s + 1.54·15-s − 1/4·16-s − 0.485·17-s − 1.41·18-s − 1.14·19-s + 0.447·20-s + 3.27·21-s + 0.852·22-s − 0.417·23-s − 1.83·24-s − 1/5·25-s + 1.37·26-s − 1.73·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11197 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−17.30355194604508, −16.69544248412029, −16.47880123802363, −15.78736162609968, −15.48467302042307, −14.71872970001793, −13.59164529057994, −13.03294065622619, −12.62621399213507, −12.22307129572691, −11.60870356674085, −10.73251051031271, −10.27521719178064, −10.00867682320660, −9.384352916495701, −8.528251728745352, −7.736943781142023, −7.064728465829807, −6.799274899613764, −5.878394085356669, −5.174340169346392, −4.655601092025473, −3.961578927392342, −3.029214703782400, −1.855407758031725, 0, 0, 0,
1.855407758031725, 3.029214703782400, 3.961578927392342, 4.655601092025473, 5.174340169346392, 5.878394085356669, 6.799274899613764, 7.064728465829807, 7.736943781142023, 8.528251728745352, 9.384352916495701, 10.00867682320660, 10.27521719178064, 10.73251051031271, 11.60870356674085, 12.22307129572691, 12.62621399213507, 13.03294065622619, 13.59164529057994, 14.71872970001793, 15.48467302042307, 15.78736162609968, 16.47880123802363, 16.69544248412029, 17.30355194604508