| L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s − 4·11-s − 6·13-s − 2·15-s + 2·17-s − 4·19-s − 21-s − 8·23-s − 25-s − 27-s + 2·29-s + 4·33-s + 2·35-s + 10·37-s + 6·39-s − 6·41-s − 4·43-s + 2·45-s + 49-s − 2·51-s − 6·53-s − 8·55-s + 4·57-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.338·35-s + 1.64·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−19.90922528213177, −19.14984933025549, −18.21453356481622, −17.97880378114661, −17.12654746213102, −16.75878332738357, −15.86724947240857, −15.08236946943927, −14.40816684431301, −13.67871015109170, −12.90054066973290, −12.28662683288716, −11.56985295135314, −10.56233382470030, −10.02545149330825, −9.563538518206812, −8.186632228453794, −7.685015150188554, −6.608076600522026, −5.765369999194802, −5.117402880349692, −4.301676378496726, −2.667923367707886, −1.883513692797836, 0,
1.883513692797836, 2.667923367707886, 4.301676378496726, 5.117402880349692, 5.765369999194802, 6.608076600522026, 7.685015150188554, 8.186632228453794, 9.563538518206812, 10.02545149330825, 10.56233382470030, 11.56985295135314, 12.28662683288716, 12.90054066973290, 13.67871015109170, 14.40816684431301, 15.08236946943927, 15.86724947240857, 16.75878332738357, 17.12654746213102, 17.97880378114661, 18.21453356481622, 19.14984933025549, 19.90922528213177
