| L(s) = 1 | + 3-s − 4·7-s − 2·9-s + 6·11-s + 13-s − 2·19-s − 4·21-s + 23-s − 5·27-s + 9·29-s − 5·31-s + 6·33-s − 2·37-s + 39-s − 9·41-s − 4·43-s − 3·47-s + 9·49-s + 6·53-s − 2·57-s + 2·61-s + 8·63-s − 10·67-s + 69-s + 3·71-s + 7·73-s − 24·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s − 2/3·9-s + 1.80·11-s + 0.277·13-s − 0.458·19-s − 0.872·21-s + 0.208·23-s − 0.962·27-s + 1.67·29-s − 0.898·31-s + 1.04·33-s − 0.328·37-s + 0.160·39-s − 1.40·41-s − 0.609·43-s − 0.437·47-s + 9/7·49-s + 0.824·53-s − 0.264·57-s + 0.256·61-s + 1.00·63-s − 1.22·67-s + 0.120·69-s + 0.356·71-s + 0.819·73-s − 2.73·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.17401257361588131674990618478, −6.61830144047848624556555914048, −6.26943130908985231694308718354, −5.43946874616994151248360207036, −4.37190988774586050003439671089, −3.55519915712606474705816051228, −3.27151498333735538174315640697, −2.32486620300571648499954408429, −1.25340285719162763913262583442, 0,
1.25340285719162763913262583442, 2.32486620300571648499954408429, 3.27151498333735538174315640697, 3.55519915712606474705816051228, 4.37190988774586050003439671089, 5.43946874616994151248360207036, 6.26943130908985231694308718354, 6.61830144047848624556555914048, 7.17401257361588131674990618478
