| L(s) = 1 | + 7-s − 4·11-s − 2·13-s + 6·17-s − 6·19-s − 2·23-s − 6·29-s + 2·31-s − 8·37-s − 12·41-s + 4·43-s − 8·47-s + 49-s − 2·53-s − 12·59-s + 10·61-s − 4·67-s + 8·71-s + 2·73-s − 4·77-s + 8·79-s + 12·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.37·19-s − 0.417·23-s − 1.11·29-s + 0.359·31-s − 1.31·37-s − 1.87·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 1.56·59-s + 1.28·61-s − 0.488·67-s + 0.949·71-s + 0.234·73-s − 0.455·77-s + 0.900·79-s + 1.27·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6852846100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6852846100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.56289301764700, −13.42412628263687, −12.65075316416637, −12.29700792842823, −11.96576253952627, −11.21271227292535, −10.69273913866487, −10.42317663112410, −9.773365858784851, −9.488468477773367, −8.558238776760090, −8.227863644645180, −7.862522865582511, −7.229093967692830, −6.785153370643595, −5.984281185658007, −5.595031072544523, −4.850139531281506, −4.775697996433149, −3.636303531217058, −3.426044230286731, −2.503729399652170, −2.022291197676772, −1.368667130855787, −0.2506007388797743,
0.2506007388797743, 1.368667130855787, 2.022291197676772, 2.503729399652170, 3.426044230286731, 3.636303531217058, 4.775697996433149, 4.850139531281506, 5.595031072544523, 5.984281185658007, 6.785153370643595, 7.229093967692830, 7.862522865582511, 8.227863644645180, 8.558238776760090, 9.488468477773367, 9.773365858784851, 10.42317663112410, 10.69273913866487, 11.21271227292535, 11.96576253952627, 12.29700792842823, 12.65075316416637, 13.42412628263687, 13.56289301764700
