| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 3·11-s − 13-s − 2·14-s + 16-s + 5·19-s + 20-s − 3·22-s − 5·23-s + 25-s + 26-s + 2·28-s + 9·29-s + 31-s − 32-s + 2·35-s + 7·37-s − 5·38-s − 40-s + 6·41-s − 11·43-s + 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 0.904·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.14·19-s + 0.223·20-s − 0.639·22-s − 1.04·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 1.67·29-s + 0.179·31-s − 0.176·32-s + 0.338·35-s + 1.15·37-s − 0.811·38-s − 0.158·40-s + 0.937·41-s − 1.67·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−12.62997991749889, −12.16867278951188, −11.87687024298948, −11.40106756881210, −11.02498088564317, −10.47240051138808, −9.954079485610123, −9.529581052534995, −9.391864837579763, −8.596339867897831, −8.248912865645878, −7.840683669062147, −7.397570837432119, −6.725705676909041, −6.378226075017711, −5.944927731770853, −5.315826183548159, −4.761139481794497, −4.365110748187140, −3.694308302124359, −2.927105645922224, −2.667309149615483, −1.744479738183631, −1.449229765265510, −0.9103767188514755, 0,
0.9103767188514755, 1.449229765265510, 1.744479738183631, 2.667309149615483, 2.927105645922224, 3.694308302124359, 4.365110748187140, 4.761139481794497, 5.315826183548159, 5.944927731770853, 6.378226075017711, 6.725705676909041, 7.397570837432119, 7.840683669062147, 8.248912865645878, 8.596339867897831, 9.391864837579763, 9.529581052534995, 9.954079485610123, 10.47240051138808, 11.02498088564317, 11.40106756881210, 11.87687024298948, 12.16867278951188, 12.62997991749889
