| L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 4·11-s − 6·13-s + 4·15-s + 4·17-s − 6·19-s − 4·23-s − 25-s + 4·27-s − 6·29-s + 4·31-s − 8·33-s − 6·37-s + 12·39-s − 4·41-s + 12·43-s − 2·45-s + 12·47-s − 8·51-s + 6·53-s − 8·55-s + 12·57-s − 6·59-s + 6·61-s + 12·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.03·15-s + 0.970·17-s − 1.37·19-s − 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.39·33-s − 0.986·37-s + 1.92·39-s − 0.624·41-s + 1.82·43-s − 0.298·45-s + 1.75·47-s − 1.12·51-s + 0.824·53-s − 1.07·55-s + 1.58·57-s − 0.781·59-s + 0.768·61-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6189072095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6189072095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.548334311913961644578260879163, −8.602967289806366862192787645068, −7.65950137952983772007530237214, −7.00517131931499789251224110851, −6.13816426608055355895026845340, −5.36896971972020825755565052761, −4.40089338508791742804895776143, −3.73481657956528417072145773521, −2.21931922241169272075459082326, −0.56712723048897687512007629436,
0.56712723048897687512007629436, 2.21931922241169272075459082326, 3.73481657956528417072145773521, 4.40089338508791742804895776143, 5.36896971972020825755565052761, 6.13816426608055355895026845340, 7.00517131931499789251224110851, 7.65950137952983772007530237214, 8.602967289806366862192787645068, 9.548334311913961644578260879163
