| L(s) = 1 | − 2·3-s − 4·7-s + 9-s + 2·11-s + 13-s − 17-s + 8·21-s + 4·23-s − 5·25-s + 4·27-s − 4·29-s + 4·31-s − 4·33-s − 12·37-s − 2·39-s + 2·41-s + 4·43-s + 9·49-s + 2·51-s − 2·53-s − 4·59-s − 4·63-s − 4·67-s − 8·69-s − 2·73-s + 10·75-s − 8·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.242·17-s + 1.74·21-s + 0.834·23-s − 25-s + 0.769·27-s − 0.742·29-s + 0.718·31-s − 0.696·33-s − 1.97·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s + 9/7·49-s + 0.280·51-s − 0.274·53-s − 0.520·59-s − 0.503·63-s − 0.488·67-s − 0.963·69-s − 0.234·73-s + 1.15·75-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−16.43887493172148, −15.89085546589773, −15.61257227625090, −14.83987608479714, −14.07342534144699, −13.48778971844338, −12.99951393445100, −12.34286507570112, −11.95249082459289, −11.40134648986838, −10.65863929017481, −10.35004748839778, −9.473585966951268, −9.162253712215179, −8.418707571018713, −7.421647674869815, −6.851696369478660, −6.297414587666473, −5.918670660378060, −5.220332839210429, −4.432710798082740, −3.596263540192502, −3.082286133147997, −1.969615427240265, −0.8325891208563326, 0,
0.8325891208563326, 1.969615427240265, 3.082286133147997, 3.596263540192502, 4.432710798082740, 5.220332839210429, 5.918670660378060, 6.297414587666473, 6.851696369478660, 7.421647674869815, 8.418707571018713, 9.162253712215179, 9.473585966951268, 10.35004748839778, 10.65863929017481, 11.40134648986838, 11.95249082459289, 12.34286507570112, 12.99951393445100, 13.48778971844338, 14.07342534144699, 14.83987608479714, 15.61257227625090, 15.89085546589773, 16.43887493172148
