L(s) = 1 | − 3·3-s − 5-s + 7-s + 6·9-s + 5·11-s − 3·13-s + 3·15-s − 17-s − 6·19-s − 3·21-s − 6·23-s + 25-s − 9·27-s − 9·29-s + 4·31-s − 15·33-s − 35-s + 2·37-s + 9·39-s − 4·41-s − 10·43-s − 6·45-s + 47-s + 49-s + 3·51-s + 4·53-s − 5·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.50·11-s − 0.832·13-s + 0.774·15-s − 0.242·17-s − 1.37·19-s − 0.654·21-s − 1.25·23-s + 1/5·25-s − 1.73·27-s − 1.67·29-s + 0.718·31-s − 2.61·33-s − 0.169·35-s + 0.328·37-s + 1.44·39-s − 0.624·41-s − 1.52·43-s − 0.894·45-s + 0.145·47-s + 1/7·49-s + 0.420·51-s + 0.549·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−10.52029628800050350235254857474, −9.716558810176534364860056345490, −8.548341637730933961960631631377, −7.32545514222645839389554035710, −6.55070763080365592509044517141, −5.79502306005565665039377390440, −4.64056350631119746271470996824, −4.00451609950028562939589775427, −1.70936955426510995854988214866, 0,
1.70936955426510995854988214866, 4.00451609950028562939589775427, 4.64056350631119746271470996824, 5.79502306005565665039377390440, 6.55070763080365592509044517141, 7.32545514222645839389554035710, 8.548341637730933961960631631377, 9.716558810176534364860056345490, 10.52029628800050350235254857474