L(s) = 1 | + 7-s − 4·11-s − 2·13-s + 2·17-s + 4·19-s − 10·29-s + 6·37-s + 6·41-s − 4·43-s − 8·47-s + 49-s − 6·53-s − 4·59-s + 10·61-s + 4·67-s + 16·71-s + 14·73-s − 4·77-s + 8·79-s + 4·83-s − 10·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.85·29-s + 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.520·59-s + 1.28·61-s + 0.488·67-s + 1.89·71-s + 1.63·73-s − 0.455·77-s + 0.900·79-s + 0.439·83-s − 1.05·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.98185922219106, −13.48831241596662, −12.93475291750475, −12.65437929234719, −12.05459786401625, −11.47362467164189, −10.99727660563270, −10.74789481596552, −9.869006323117043, −9.588699356498321, −9.275925174928096, −8.274748200400645, −7.830558462511191, −7.786999477221252, −6.955950457147894, −6.494109161548620, −5.565523163240397, −5.357341085903523, −4.930445219616171, −4.116947295083129, −3.562538282184670, −2.878101846015721, −2.338427303907269, −1.680642829502003, −0.8323018351488837, 0,
0.8323018351488837, 1.680642829502003, 2.338427303907269, 2.878101846015721, 3.562538282184670, 4.116947295083129, 4.930445219616171, 5.357341085903523, 5.565523163240397, 6.494109161548620, 6.955950457147894, 7.786999477221252, 7.830558462511191, 8.274748200400645, 9.275925174928096, 9.588699356498321, 9.869006323117043, 10.74789481596552, 10.99727660563270, 11.47362467164189, 12.05459786401625, 12.65437929234719, 12.93475291750475, 13.48831241596662, 13.98185922219106