| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s + 11-s + 12-s − 13-s − 2·14-s + 16-s + 17-s + 18-s + 4·19-s − 2·21-s + 22-s − 6·23-s + 24-s − 26-s + 27-s − 2·28-s − 8·29-s + 2·31-s + 32-s + 33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.377·28-s − 1.48·29-s + 0.359·31-s + 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.70795871602905, −12.39656983234945, −11.98034631642777, −11.38862730803410, −11.16872971308053, −10.32277364849026, −9.956352353234983, −9.696987211519737, −9.180666687794195, −8.636991909507069, −8.025695189303244, −7.719596386046927, −7.121729015438563, −6.748580423909924, −6.240520289163934, −5.749192906611291, −5.175543292729292, −4.839204846963446, −3.963283908063718, −3.680837883815359, −3.361418983355558, −2.682649061227621, −2.117901626850728, −1.660486984780540, −0.8410061457994798, 0,
0.8410061457994798, 1.660486984780540, 2.117901626850728, 2.682649061227621, 3.361418983355558, 3.680837883815359, 3.963283908063718, 4.839204846963446, 5.175543292729292, 5.749192906611291, 6.240520289163934, 6.748580423909924, 7.121729015438563, 7.719596386046927, 8.025695189303244, 8.636991909507069, 9.180666687794195, 9.696987211519737, 9.956352353234983, 10.32277364849026, 11.16872971308053, 11.38862730803410, 11.98034631642777, 12.39656983234945, 12.70795871602905
