| L(s) = 1 | + 3-s + 9-s + 4·11-s − 2·13-s − 2·17-s − 4·19-s − 8·23-s + 27-s − 6·29-s − 8·31-s + 4·33-s + 6·37-s − 2·39-s − 6·41-s − 4·43-s − 7·49-s − 2·51-s − 2·53-s − 4·57-s + 4·59-s + 2·61-s + 4·67-s − 8·69-s − 8·71-s − 10·73-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 49-s − 0.280·51-s − 0.274·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.488·67-s − 0.963·69-s − 0.949·71-s − 1.17·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.953704755279657121335115603175, −7.25136384633542617765663547930, −6.50923348756050362748698725466, −5.87866003990815751467424375923, −4.79956865275231776293703222232, −4.02593153952443544832427859826, −3.48889300562368399939117276327, −2.23734692694206799493488443258, −1.66440253308003017822578074122, 0,
1.66440253308003017822578074122, 2.23734692694206799493488443258, 3.48889300562368399939117276327, 4.02593153952443544832427859826, 4.79956865275231776293703222232, 5.87866003990815751467424375923, 6.50923348756050362748698725466, 7.25136384633542617765663547930, 7.953704755279657121335115603175
