L(s) = 1 | + 5-s + 4·7-s + 4·13-s − 6·17-s − 2·19-s + 25-s − 4·31-s + 4·35-s − 10·37-s + 4·43-s − 12·47-s + 9·49-s − 6·53-s − 12·59-s + 10·61-s + 4·65-s − 4·67-s − 8·73-s + 10·79-s − 6·83-s − 6·85-s + 6·89-s + 16·91-s − 2·95-s − 10·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.10·13-s − 1.45·17-s − 0.458·19-s + 1/5·25-s − 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s − 0.936·73-s + 1.12·79-s − 0.658·83-s − 0.650·85-s + 0.635·89-s + 1.67·91-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.80412187349826, −15.18231953504560, −14.77624847466948, −14.10589569608724, −13.76449429925498, −13.17142827294494, −12.65799384368991, −11.89461712370386, −11.29412005929072, −10.88194521429956, −10.60998918430196, −9.689622160503713, −8.973603002852378, −8.596742858824746, −8.120072588016054, −7.403333680531554, −6.658703689231612, −6.197749556890042, −5.374424794742147, −4.880559067993424, −4.251653625691743, −3.570407754172462, −2.570179849855544, −1.749405339285517, −1.420129610281989, 0,
1.420129610281989, 1.749405339285517, 2.570179849855544, 3.570407754172462, 4.251653625691743, 4.880559067993424, 5.374424794742147, 6.197749556890042, 6.658703689231612, 7.403333680531554, 8.120072588016054, 8.596742858824746, 8.973603002852378, 9.689622160503713, 10.60998918430196, 10.88194521429956, 11.29412005929072, 11.89461712370386, 12.65799384368991, 13.17142827294494, 13.76449429925498, 14.10589569608724, 14.77624847466948, 15.18231953504560, 15.80412187349826