| L(s) = 1 | + 2·7-s + 2·11-s − 13-s + 2·17-s + 4·19-s − 23-s + 29-s + 2·31-s − 12·41-s + 5·43-s − 4·47-s − 3·49-s + 9·53-s − 8·59-s − 7·61-s − 14·67-s + 6·73-s + 4·77-s + 15·79-s − 4·83-s + 18·89-s − 2·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.603·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s − 0.208·23-s + 0.185·29-s + 0.359·31-s − 1.87·41-s + 0.762·43-s − 0.583·47-s − 3/7·49-s + 1.23·53-s − 1.04·59-s − 0.896·61-s − 1.71·67-s + 0.702·73-s + 0.455·77-s + 1.68·79-s − 0.439·83-s + 1.90·89-s − 0.209·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.48636182914643, −12.92787289246484, −12.12618265189867, −11.91806058085323, −11.76340386288832, −10.92020357717626, −10.59922449001965, −10.10729595575318, −9.377762533964203, −9.301476671963999, −8.536636223039097, −8.008058786225759, −7.747540095921236, −7.114112895428877, −6.594143992336870, −6.139649857845898, −5.327473360135550, −5.179086734352793, −4.465306138843620, −4.002443563913665, −3.282271298235831, −2.889547671577535, −2.027886497454010, −1.491187355411010, −0.9612378625244717, 0,
0.9612378625244717, 1.491187355411010, 2.027886497454010, 2.889547671577535, 3.282271298235831, 4.002443563913665, 4.465306138843620, 5.179086734352793, 5.327473360135550, 6.139649857845898, 6.594143992336870, 7.114112895428877, 7.747540095921236, 8.008058786225759, 8.536636223039097, 9.301476671963999, 9.377762533964203, 10.10729595575318, 10.59922449001965, 10.92020357717626, 11.76340386288832, 11.91806058085323, 12.12618265189867, 12.92787289246484, 13.48636182914643
