| L(s) = 1 | − 3-s − 4·5-s − 2·9-s − 3·11-s + 2·13-s + 4·15-s + 2·17-s − 6·23-s + 11·25-s + 5·27-s − 4·29-s + 10·31-s + 3·33-s + 2·37-s − 2·39-s + 9·41-s + 4·43-s + 8·45-s + 12·47-s − 7·49-s − 2·51-s − 2·53-s + 12·55-s + 59-s − 8·61-s − 8·65-s − 9·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 2/3·9-s − 0.904·11-s + 0.554·13-s + 1.03·15-s + 0.485·17-s − 1.25·23-s + 11/5·25-s + 0.962·27-s − 0.742·29-s + 1.79·31-s + 0.522·33-s + 0.328·37-s − 0.320·39-s + 1.40·41-s + 0.609·43-s + 1.19·45-s + 1.75·47-s − 49-s − 0.280·51-s − 0.274·53-s + 1.61·55-s + 0.130·59-s − 1.02·61-s − 0.992·65-s − 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.84291285553569302803646666567, −7.25281256893503607163488926699, −6.18133217332434823974698514600, −5.72056544906938218909518824058, −4.70361893560692538945006993689, −4.16988049041125598647554401476, −3.30209644061830425490225828028, −2.57233604129214373443319462753, −0.900922910872936613997172410594, 0,
0.900922910872936613997172410594, 2.57233604129214373443319462753, 3.30209644061830425490225828028, 4.16988049041125598647554401476, 4.70361893560692538945006993689, 5.72056544906938218909518824058, 6.18133217332434823974698514600, 7.25281256893503607163488926699, 7.84291285553569302803646666567
