L(s) = 1 | − 5-s + 7-s + 4·11-s + 23-s + 25-s − 2·29-s + 4·31-s − 35-s + 2·37-s − 4·43-s − 8·47-s + 49-s − 2·53-s − 4·55-s − 10·59-s − 12·61-s + 4·67-s − 2·73-s + 4·77-s + 2·79-s + 14·83-s + 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.208·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.539·55-s − 1.30·59-s − 1.53·61-s + 0.488·67-s − 0.234·73-s + 0.455·77-s + 0.225·79-s + 1.53·83-s + 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.79185081075818, −13.47303953398619, −12.83674486139525, −12.16675053197130, −12.04033208237107, −11.38544168080339, −11.05255050903205, −10.55247758034820, −9.812298526920975, −9.475948925506163, −8.893577306265709, −8.450286632171527, −7.899186737550057, −7.428051774670369, −6.881288948668822, −6.215140030029347, −6.046453637603531, −4.943310761668400, −4.794238129988509, −4.095057562462538, −3.508201817030316, −3.064530604162012, −2.178700289291983, −1.529746221625468, −0.9439102523328816, 0,
0.9439102523328816, 1.529746221625468, 2.178700289291983, 3.064530604162012, 3.508201817030316, 4.095057562462538, 4.794238129988509, 4.943310761668400, 6.046453637603531, 6.215140030029347, 6.881288948668822, 7.428051774670369, 7.899186737550057, 8.450286632171527, 8.893577306265709, 9.475948925506163, 9.812298526920975, 10.55247758034820, 11.05255050903205, 11.38544168080339, 12.04033208237107, 12.16675053197130, 12.83674486139525, 13.47303953398619, 13.79185081075818