| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s − 11-s + 6·13-s + 4·14-s + 16-s − 17-s − 2·19-s − 20-s + 22-s + 8·23-s + 25-s − 6·26-s − 4·28-s − 2·29-s − 32-s + 34-s + 4·35-s − 10·37-s + 2·38-s + 40-s − 8·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.458·19-s − 0.223·20-s + 0.213·22-s + 1.66·23-s + 1/5·25-s − 1.17·26-s − 0.755·28-s − 0.371·29-s − 0.176·32-s + 0.171·34-s + 0.676·35-s − 1.64·37-s + 0.324·38-s + 0.158·40-s − 1.21·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.10939330844251, −15.85756193027283, −15.19506246307947, −14.85066105773647, −13.73214084461471, −13.33827370454400, −12.86812505054256, −12.34135376335811, −11.51896768969376, −11.09851538178977, −10.42705788288338, −10.10980719325743, −9.150488357623613, −8.883567075091364, −8.374820084634436, −7.551053744349976, −6.766523717695929, −6.616405597033698, −5.819115923106963, −5.097287290294621, −4.035991741597314, −3.363723948800552, −3.014508636657700, −1.888343333204660, −0.9026535965453123, 0,
0.9026535965453123, 1.888343333204660, 3.014508636657700, 3.363723948800552, 4.035991741597314, 5.097287290294621, 5.819115923106963, 6.616405597033698, 6.766523717695929, 7.551053744349976, 8.374820084634436, 8.883567075091364, 9.150488357623613, 10.10980719325743, 10.42705788288338, 11.09851538178977, 11.51896768969376, 12.34135376335811, 12.86812505054256, 13.33827370454400, 13.73214084461471, 14.85066105773647, 15.19506246307947, 15.85756193027283, 16.10939330844251
