| L(s) = 1 | + 7-s − 3·9-s + 11-s − 13-s + 5·19-s + 23-s − 5·29-s + 2·31-s + 4·37-s − 5·41-s − 9·43-s − 6·47-s − 6·49-s − 2·53-s − 8·59-s − 8·61-s − 3·63-s + 8·67-s + 10·71-s + 3·73-s + 77-s + 3·79-s + 9·81-s + 3·83-s + 10·89-s − 91-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 0.301·11-s − 0.277·13-s + 1.14·19-s + 0.208·23-s − 0.928·29-s + 0.359·31-s + 0.657·37-s − 0.780·41-s − 1.37·43-s − 0.875·47-s − 6/7·49-s − 0.274·53-s − 1.04·59-s − 1.02·61-s − 0.377·63-s + 0.977·67-s + 1.18·71-s + 0.351·73-s + 0.113·77-s + 0.337·79-s + 81-s + 0.329·83-s + 1.05·89-s − 0.104·91-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 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
−7.52182919983062274835355996980, −6.60423240068529020085303208340, −6.07209503282695111135719683981, −5.11954952754445966688807183650, −4.88902726553666115997931683381, −3.64695327025583856949172622415, −3.14625447518025092438654521785, −2.19079632817127595853391594771, −1.25205173106337267340854487755, 0,
1.25205173106337267340854487755, 2.19079632817127595853391594771, 3.14625447518025092438654521785, 3.64695327025583856949172622415, 4.88902726553666115997931683381, 5.11954952754445966688807183650, 6.07209503282695111135719683981, 6.60423240068529020085303208340, 7.52182919983062274835355996980
