| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s + 9-s − 3·11-s − 12-s − 3·13-s + 3·14-s + 16-s − 4·17-s + 18-s − 3·19-s − 3·21-s − 3·22-s − 23-s − 24-s − 3·26-s − 27-s + 3·28-s + 3·29-s − 10·31-s + 32-s + 3·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s − 0.832·13-s + 0.801·14-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.688·19-s − 0.654·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s − 0.588·26-s − 0.192·27-s + 0.566·28-s + 0.557·29-s − 1.79·31-s + 0.176·32-s + 0.522·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.154459911464843641240378090210, −7.25319430573909960362680373091, −6.78411513804883783711084072693, −5.69232326633719283569498398524, −5.09453579990225833632965450691, −4.62571344330038805628278337051, −3.69025613137568425396049248589, −2.40987911924220110783420611976, −1.74358696081969393863801582349, 0,
1.74358696081969393863801582349, 2.40987911924220110783420611976, 3.69025613137568425396049248589, 4.62571344330038805628278337051, 5.09453579990225833632965450691, 5.69232326633719283569498398524, 6.78411513804883783711084072693, 7.25319430573909960362680373091, 8.154459911464843641240378090210
