L(s) = 1 | + 2·2-s + 3·3-s + 2·4-s − 2·5-s + 6·6-s + 3·7-s + 6·9-s − 4·10-s + 11-s + 6·12-s + 13-s + 6·14-s − 6·15-s − 4·16-s + 2·17-s + 12·18-s − 4·19-s − 4·20-s + 9·21-s + 2·22-s − 25-s + 2·26-s + 9·27-s + 6·28-s − 12·30-s − 8·32-s + 3·33-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s + 1.13·7-s + 2·9-s − 1.26·10-s + 0.301·11-s + 1.73·12-s + 0.277·13-s + 1.60·14-s − 1.54·15-s − 16-s + 0.485·17-s + 2.82·18-s − 0.917·19-s − 0.894·20-s + 1.96·21-s + 0.426·22-s − 1/5·25-s + 0.392·26-s + 1.73·27-s + 1.13·28-s − 2.19·30-s − 1.41·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.926199236\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.926199236\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.188319790101530921753204184271, −8.616077764144528028615874416326, −7.87477791859352406459688530912, −7.30582572005395966038296105626, −6.16102246163409103474548988436, −4.92291875338996393170858646518, −4.17112318724767747236517698099, −3.69086947647099089907189542872, −2.75577143583230419319198481044, −1.73705984985495929063658157655,
1.73705984985495929063658157655, 2.75577143583230419319198481044, 3.69086947647099089907189542872, 4.17112318724767747236517698099, 4.92291875338996393170858646518, 6.16102246163409103474548988436, 7.30582572005395966038296105626, 7.87477791859352406459688530912, 8.616077764144528028615874416326, 9.188319790101530921753204184271