| L(s) = 1 | − 3-s + 4·7-s + 9-s − 2·13-s − 2·17-s + 2·19-s − 4·21-s + 6·23-s − 5·25-s − 27-s + 2·29-s − 8·31-s + 2·37-s + 2·39-s − 6·41-s + 2·43-s + 6·47-s + 9·49-s + 2·51-s − 4·53-s − 2·57-s − 4·59-s − 10·61-s + 4·63-s − 12·67-s − 6·69-s + 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.485·17-s + 0.458·19-s − 0.872·21-s + 1.25·23-s − 25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.320·39-s − 0.937·41-s + 0.304·43-s + 0.875·47-s + 9/7·49-s + 0.280·51-s − 0.549·53-s − 0.264·57-s − 0.520·59-s − 1.28·61-s + 0.503·63-s − 1.46·67-s − 0.722·69-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.55110442736474, −15.19779586710256, −14.78759471852041, −14.04324923379285, −13.70598807344281, −13.01604889901412, −12.25102539342836, −12.00853890555053, −11.22791367150874, −10.95633010956027, −10.52436512522570, −9.539556858294243, −9.252948951288387, −8.411387053095833, −7.877996865325981, −7.316590407221906, −6.835870428638116, −5.936680141112220, −5.355541542926958, −4.868131169692232, −4.361213720443243, −3.546110133594277, −2.570012549459206, −1.794960811055191, −1.154924996497900, 0,
1.154924996497900, 1.794960811055191, 2.570012549459206, 3.546110133594277, 4.361213720443243, 4.868131169692232, 5.355541542926958, 5.936680141112220, 6.835870428638116, 7.316590407221906, 7.877996865325981, 8.411387053095833, 9.252948951288387, 9.539556858294243, 10.52436512522570, 10.95633010956027, 11.22791367150874, 12.00853890555053, 12.25102539342836, 13.01604889901412, 13.70598807344281, 14.04324923379285, 14.78759471852041, 15.19779586710256, 15.55110442736474
