| L(s) = 1 | + 5-s − 4·7-s + 11-s − 4·13-s − 2·17-s + 6·19-s − 6·23-s + 25-s + 6·29-s − 4·35-s + 10·37-s + 6·41-s + 10·43-s − 6·47-s + 9·49-s + 10·53-s + 55-s − 4·65-s − 12·67-s − 14·71-s − 2·73-s − 4·77-s − 8·79-s − 4·83-s − 2·85-s + 16·91-s + 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 0.301·11-s − 1.10·13-s − 0.485·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.676·35-s + 1.64·37-s + 0.937·41-s + 1.52·43-s − 0.875·47-s + 9/7·49-s + 1.37·53-s + 0.134·55-s − 0.496·65-s − 1.46·67-s − 1.66·71-s − 0.234·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-s − 0.216·85-s + 1.67·91-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.36065042187922316871189737108, −6.82274933461576479982463644147, −6.02803118076422837225082724609, −5.67448842577354044735564145530, −4.56940328893734604313903275505, −3.95553657034785344772335686804, −2.82444339529599280030323006962, −2.58244783409107657486909735879, −1.17134687290941581926893957198, 0,
1.17134687290941581926893957198, 2.58244783409107657486909735879, 2.82444339529599280030323006962, 3.95553657034785344772335686804, 4.56940328893734604313903275505, 5.67448842577354044735564145530, 6.02803118076422837225082724609, 6.82274933461576479982463644147, 7.36065042187922316871189737108
