| L(s) = 1 | − 3-s − 3·5-s + 3·7-s + 9-s − 2·11-s + 2·13-s + 3·15-s − 4·19-s − 3·21-s − 5·23-s + 4·25-s − 27-s − 4·29-s + 5·31-s + 2·33-s − 9·35-s − 11·37-s − 2·39-s − 5·41-s − 5·43-s − 3·45-s − 8·47-s + 2·49-s + 53-s + 6·55-s + 4·57-s + 9·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.774·15-s − 0.917·19-s − 0.654·21-s − 1.04·23-s + 4/5·25-s − 0.192·27-s − 0.742·29-s + 0.898·31-s + 0.348·33-s − 1.52·35-s − 1.80·37-s − 0.320·39-s − 0.780·41-s − 0.762·43-s − 0.447·45-s − 1.16·47-s + 2/7·49-s + 0.137·53-s + 0.809·55-s + 0.529·57-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−10.15868869792210616464939168318, −8.621750975497364773066854866729, −8.146136605098107917617637872208, −7.39926385220760222612819568905, −6.35864929001434414604203087965, −5.19473446533380084306467364703, −4.43849216334823672315538044895, −3.52236372066033986255555688740, −1.78029060925712868391460402926, 0,
1.78029060925712868391460402926, 3.52236372066033986255555688740, 4.43849216334823672315538044895, 5.19473446533380084306467364703, 6.35864929001434414604203087965, 7.39926385220760222612819568905, 8.146136605098107917617637872208, 8.621750975497364773066854866729, 10.15868869792210616464939168318
