| L(s) = 1 | − 2·3-s + 5-s − 4·7-s + 9-s + 4·11-s − 2·15-s + 6·17-s + 19-s + 8·21-s − 8·23-s + 25-s + 4·27-s − 6·29-s + 8·31-s − 8·33-s − 4·35-s − 8·37-s − 2·41-s + 45-s − 12·47-s + 9·49-s − 12·51-s + 4·53-s + 4·55-s − 2·57-s − 8·59-s − 14·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s + 1.45·17-s + 0.229·19-s + 1.74·21-s − 1.66·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 1.43·31-s − 1.39·33-s − 0.676·35-s − 1.31·37-s − 0.312·41-s + 0.149·45-s − 1.75·47-s + 9/7·49-s − 1.68·51-s + 0.549·53-s + 0.539·55-s − 0.264·57-s − 1.04·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−9.414389171153941052152571176806, −8.306558588442832701284063292788, −7.15092239567899718438965748252, −6.26604533373139617436495529729, −6.07553649243771927211326383485, −5.14236160489505260986626044288, −3.89039366794407092271004025254, −3.07425032684048770058081362284, −1.43164889163469816022501322949, 0,
1.43164889163469816022501322949, 3.07425032684048770058081362284, 3.89039366794407092271004025254, 5.14236160489505260986626044288, 6.07553649243771927211326383485, 6.26604533373139617436495529729, 7.15092239567899718438965748252, 8.306558588442832701284063292788, 9.414389171153941052152571176806
