| L(s) = 1 | − 5-s − 7-s + 11-s + 13-s − 17-s + 7·19-s − 3·23-s + 25-s + 8·29-s + 5·31-s + 35-s + 5·37-s + 10·41-s − 2·43-s − 6·49-s − 10·53-s − 55-s − 4·59-s − 9·61-s − 65-s + 9·67-s + 4·71-s − 10·73-s − 77-s + 2·79-s + 3·83-s + 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.301·11-s + 0.277·13-s − 0.242·17-s + 1.60·19-s − 0.625·23-s + 1/5·25-s + 1.48·29-s + 0.898·31-s + 0.169·35-s + 0.821·37-s + 1.56·41-s − 0.304·43-s − 6/7·49-s − 1.37·53-s − 0.134·55-s − 0.520·59-s − 1.15·61-s − 0.124·65-s + 1.09·67-s + 0.474·71-s − 1.17·73-s − 0.113·77-s + 0.225·79-s + 0.329·83-s + 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.70196376272442, −13.26146841018232, −12.63574125306827, −12.21751927523406, −11.82355188466342, −11.30458413891411, −10.92378915771547, −10.25589548186925, −9.748486430710376, −9.427526507588829, −8.864473359655869, −8.205087958106410, −7.841192545487425, −7.399914851065399, −6.689527064902308, −6.240437606779635, −5.900979506512873, −4.979869574054220, −4.679515645486962, −4.027932572807203, −3.418218619928362, −2.927364287366618, −2.389299416975988, −1.341140842802013, −0.9382228852952926, 0,
0.9382228852952926, 1.341140842802013, 2.389299416975988, 2.927364287366618, 3.418218619928362, 4.027932572807203, 4.679515645486962, 4.979869574054220, 5.900979506512873, 6.240437606779635, 6.689527064902308, 7.399914851065399, 7.841192545487425, 8.205087958106410, 8.864473359655869, 9.427526507588829, 9.748486430710376, 10.25589548186925, 10.92378915771547, 11.30458413891411, 11.82355188466342, 12.21751927523406, 12.63574125306827, 13.26146841018232, 13.70196376272442
