| L(s) = 1 | − 3-s + 2·7-s + 9-s + 11-s + 8·17-s − 8·19-s − 2·21-s − 4·23-s − 27-s − 6·29-s − 33-s − 6·37-s − 2·41-s − 2·43-s + 4·47-s − 3·49-s − 8·51-s + 2·53-s + 8·57-s − 12·59-s − 6·61-s + 2·63-s − 8·67-s + 4·69-s + 8·73-s + 2·77-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.94·17-s − 1.83·19-s − 0.436·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.174·33-s − 0.986·37-s − 0.312·41-s − 0.304·43-s + 0.583·47-s − 3/7·49-s − 1.12·51-s + 0.274·53-s + 1.05·57-s − 1.56·59-s − 0.768·61-s + 0.251·63-s − 0.977·67-s + 0.481·69-s + 0.936·73-s + 0.227·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.74895224861211934344212559298, −6.91800319377342600488534381831, −6.11471483375634912234659726615, −5.58775197583899428795747750682, −4.82182483302052790566799657483, −4.08116482414946065718919477576, −3.33250411529309078099703888093, −2.04273586585982124519344590835, −1.36386887048717306307694099362, 0,
1.36386887048717306307694099362, 2.04273586585982124519344590835, 3.33250411529309078099703888093, 4.08116482414946065718919477576, 4.82182483302052790566799657483, 5.58775197583899428795747750682, 6.11471483375634912234659726615, 6.91800319377342600488534381831, 7.74895224861211934344212559298
