| L(s) = 1 | − 2·3-s − 7-s + 9-s − 4·11-s + 6·17-s + 6·19-s + 2·21-s + 23-s − 5·25-s + 4·27-s + 10·29-s − 4·31-s + 8·33-s − 2·37-s − 10·41-s + 4·43-s − 12·47-s + 49-s − 12·51-s − 6·53-s − 12·57-s + 2·59-s − 63-s − 2·69-s + 8·71-s − 6·73-s + 10·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.45·17-s + 1.37·19-s + 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s + 1.39·33-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 1.68·51-s − 0.824·53-s − 1.58·57-s + 0.260·59-s − 0.125·63-s − 0.240·69-s + 0.949·71-s − 0.702·73-s + 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.301744712687195431760292309539, −7.74988710039643353360575497141, −6.85177319222593067310879867892, −6.08177121463097384563662103024, −5.24601487899152555441015567209, −5.03388474470559153479602939879, −3.54598166631464413056339917558, −2.79551625971028242736398494921, −1.23146513560978814112389724417, 0,
1.23146513560978814112389724417, 2.79551625971028242736398494921, 3.54598166631464413056339917558, 5.03388474470559153479602939879, 5.24601487899152555441015567209, 6.08177121463097384563662103024, 6.85177319222593067310879867892, 7.74988710039643353360575497141, 8.301744712687195431760292309539
