| L(s) = 1 | − 2-s − 2·3-s + 4-s − 2·5-s + 2·6-s − 8-s + 9-s + 2·10-s + 11-s − 2·12-s + 4·13-s + 4·15-s + 16-s − 18-s − 4·19-s − 2·20-s − 22-s + 4·23-s + 2·24-s − 25-s − 4·26-s + 4·27-s + 2·29-s − 4·30-s + 10·31-s − 32-s − 2·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.10·13-s + 1.03·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.213·22-s + 0.834·23-s + 0.408·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.371·29-s − 0.730·30-s + 1.79·31-s − 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.512353895122736054642994261496, −8.438279870467851105214692730660, −8.059886585841006026613889387391, −6.63986065044552740769645726528, −6.45975539812673080939370676867, −5.23639126138259380433417465369, −4.24221631118951556259235272721, −3.09515527693911842740827286057, −1.30846072392587640239816399795, 0,
1.30846072392587640239816399795, 3.09515527693911842740827286057, 4.24221631118951556259235272721, 5.23639126138259380433417465369, 6.45975539812673080939370676867, 6.63986065044552740769645726528, 8.059886585841006026613889387391, 8.438279870467851105214692730660, 9.512353895122736054642994261496
