| L(s) = 1 | + 7-s − 4·11-s + 3·13-s − 4·17-s + 19-s + 8·29-s − 31-s − 2·37-s − 2·41-s − 11·43-s + 2·47-s − 6·49-s − 10·53-s − 6·59-s + 11·61-s + 9·67-s + 6·71-s + 14·73-s − 4·77-s − 16·79-s − 2·83-s + 3·91-s + 11·97-s − 14·101-s + 8·103-s − 6·107-s − 11·109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s + 0.832·13-s − 0.970·17-s + 0.229·19-s + 1.48·29-s − 0.179·31-s − 0.328·37-s − 0.312·41-s − 1.67·43-s + 0.291·47-s − 6/7·49-s − 1.37·53-s − 0.781·59-s + 1.40·61-s + 1.09·67-s + 0.712·71-s + 1.63·73-s − 0.455·77-s − 1.80·79-s − 0.219·83-s + 0.314·91-s + 1.11·97-s − 1.39·101-s + 0.788·103-s − 0.580·107-s − 1.05·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.72976433612443188588237557125, −6.73492340398703681227628022001, −6.34720055628381170877224491641, −5.23399467997362960383786705628, −4.92884766890271290138433044796, −3.95880548395019386369268689291, −3.09766889298119750243213890847, −2.29421886983463011439159285456, −1.31752447688370921145803866253, 0,
1.31752447688370921145803866253, 2.29421886983463011439159285456, 3.09766889298119750243213890847, 3.95880548395019386369268689291, 4.92884766890271290138433044796, 5.23399467997362960383786705628, 6.34720055628381170877224491641, 6.73492340398703681227628022001, 7.72976433612443188588237557125
