| L(s) = 1 | + 4·7-s − 4·11-s + 6·13-s − 2·17-s − 4·23-s − 5·25-s − 10·37-s − 6·41-s − 4·43-s + 6·47-s + 9·49-s + 6·53-s − 4·59-s − 12·61-s − 4·67-s − 6·71-s − 14·73-s − 16·77-s − 8·79-s + 4·83-s + 6·89-s + 24·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.834·23-s − 25-s − 1.64·37-s − 0.937·41-s − 0.609·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 0.520·59-s − 1.53·61-s − 0.488·67-s − 0.712·71-s − 1.63·73-s − 1.82·77-s − 0.900·79-s + 0.439·83-s + 0.635·89-s + 2.51·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 139 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−17.01958094182719, −16.20871424346465, −15.67667728923920, −15.35790398617534, −14.67746023634055, −13.81109547993785, −13.65281002246826, −13.09767732525274, −12.03795550000705, −11.77379054449980, −10.99154653785541, −10.58043110415776, −10.12982463858369, −8.945206188485247, −8.599618590418927, −7.962895847071679, −7.551624422509277, −6.623438910282091, −5.770681522899481, −5.364930593789499, −4.522642893235182, −3.924081601870843, −3.004242835416763, −1.937976221742638, −1.457528446621464, 0,
1.457528446621464, 1.937976221742638, 3.004242835416763, 3.924081601870843, 4.522642893235182, 5.364930593789499, 5.770681522899481, 6.623438910282091, 7.551624422509277, 7.962895847071679, 8.599618590418927, 8.945206188485247, 10.12982463858369, 10.58043110415776, 10.99154653785541, 11.77379054449980, 12.03795550000705, 13.09767732525274, 13.65281002246826, 13.81109547993785, 14.67746023634055, 15.35790398617534, 15.67667728923920, 16.20871424346465, 17.01958094182719
