| L(s) = 1 | + 3-s − 2.77·5-s + 0.556·7-s + 9-s + 3.82·11-s − 0.0549·13-s − 2.77·15-s − 2.00·17-s − 2.76·19-s + 0.556·21-s + 23-s + 2.69·25-s + 27-s + 29-s − 3.02·31-s + 3.82·33-s − 1.54·35-s + 1.22·37-s − 0.0549·39-s − 6.65·41-s − 4.40·43-s − 2.77·45-s − 4.00·47-s − 6.69·49-s − 2.00·51-s − 3.40·53-s − 10.6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.24·5-s + 0.210·7-s + 0.333·9-s + 1.15·11-s − 0.0152·13-s − 0.716·15-s − 0.486·17-s − 0.634·19-s + 0.121·21-s + 0.208·23-s + 0.539·25-s + 0.192·27-s + 0.185·29-s − 0.544·31-s + 0.666·33-s − 0.260·35-s + 0.200·37-s − 0.00880·39-s − 1.03·41-s − 0.671·43-s − 0.413·45-s − 0.584·47-s − 0.955·49-s − 0.281·51-s − 0.467·53-s − 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 - 0.556T + 7T^{2} \) |
| 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 + 0.0549T + 13T^{2} \) |
| 17 | \( 1 + 2.00T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 31 | \( 1 + 3.02T + 31T^{2} \) |
| 37 | \( 1 - 1.22T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 + 4.40T + 43T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 + 0.00393T + 67T^{2} \) |
| 71 | \( 1 + 3.59T + 71T^{2} \) |
| 73 | \( 1 - 7.82T + 73T^{2} \) |
| 79 | \( 1 - 3.03T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 9.71T + 97T^{2} \) |
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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
−7.58388799353216636338528020802, −6.81039609155612411905967265899, −6.43118578706430900935259776431, −5.19574598137202812016696077186, −4.46533990411834888934287106678, −3.82859924588550259795994090635, −3.34686455140572478766511708091, −2.23949710425103708705234735818, −1.30161477152882774194915738861, 0,
1.30161477152882774194915738861, 2.23949710425103708705234735818, 3.34686455140572478766511708091, 3.82859924588550259795994090635, 4.46533990411834888934287106678, 5.19574598137202812016696077186, 6.43118578706430900935259776431, 6.81039609155612411905967265899, 7.58388799353216636338528020802
