| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s − 11-s − 6·13-s − 2·14-s + 16-s − 17-s + 4·19-s − 20-s + 22-s + 8·23-s + 25-s + 6·26-s + 2·28-s + 10·29-s − 32-s + 34-s − 2·35-s − 10·37-s − 4·38-s + 40-s − 6·41-s + 4·43-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 − 0.301·11-s − 1.66·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s − 0.223·20-s + 0.213·22-s + 1.66·23-s + 1/5·25-s + 1.17·26-s + 0.377·28-s + 1.85·29-s − 0.176·32-s + 0.171·34-s − 0.338·35-s − 1.64·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s + 0.609·43-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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−16.17941447135841, −15.61551178544687, −15.12733628637217, −14.71396167337936, −14.01466433395735, −13.53417915140868, −12.58130441477421, −12.11754646222091, −11.79027248385481, −11.01980659898506, −10.57384561052478, −9.970666527715521, −9.384030592191781, −8.670018463787006, −8.277710507911281, −7.430104590577079, −7.235914556845833, −6.574169462788319, −5.524612311554946, −4.825688426333736, −4.629717432549917, −3.234424676005865, −2.850221575315606, −1.891898882855453, −1.032665740211721, 0,
1.032665740211721, 1.891898882855453, 2.850221575315606, 3.234424676005865, 4.629717432549917, 4.825688426333736, 5.524612311554946, 6.574169462788319, 7.235914556845833, 7.430104590577079, 8.277710507911281, 8.670018463787006, 9.384030592191781, 9.970666527715521, 10.57384561052478, 11.01980659898506, 11.79027248385481, 12.11754646222091, 12.58130441477421, 13.53417915140868, 14.01466433395735, 14.71396167337936, 15.12733628637217, 15.61551178544687, 16.17941447135841
