| L(s) = 1 | − 3·9-s − 3·11-s − 5·13-s + 2·17-s − 5·19-s − 7·23-s + 4·29-s + 2·31-s − 37-s − 3·41-s − 2·43-s + 7·47-s − 9·53-s − 4·59-s + 6·61-s − 2·67-s − 6·71-s + 16·73-s + 14·79-s + 9·81-s − 6·83-s − 2·89-s + 12·97-s + 9·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 9-s − 0.904·11-s − 1.38·13-s + 0.485·17-s − 1.14·19-s − 1.45·23-s + 0.742·29-s + 0.359·31-s − 0.164·37-s − 0.468·41-s − 0.304·43-s + 1.02·47-s − 1.23·53-s − 0.520·59-s + 0.768·61-s − 0.244·67-s − 0.712·71-s + 1.87·73-s + 1.57·79-s + 81-s − 0.658·83-s − 0.211·89-s + 1.21·97-s + 0.904·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.18507385084618, −13.96877540787324, −13.26812947645242, −12.68575949533869, −12.17390135682884, −11.99348552764213, −11.25690891519393, −10.72367527292074, −10.21066168087664, −9.886292209635589, −9.252798547918125, −8.559053308312283, −8.172805520982063, −7.742290158034666, −7.186002991923640, −6.391990550365510, −6.054637133375505, −5.366688123633344, −4.885295450736356, −4.391234750334183, −3.543955763814092, −2.961775552004383, −2.288076692348025, −1.981646921503662, −0.6480545962837746, 0,
0.6480545962837746, 1.981646921503662, 2.288076692348025, 2.961775552004383, 3.543955763814092, 4.391234750334183, 4.885295450736356, 5.366688123633344, 6.054637133375505, 6.391990550365510, 7.186002991923640, 7.742290158034666, 8.172805520982063, 8.559053308312283, 9.252798547918125, 9.886292209635589, 10.21066168087664, 10.72367527292074, 11.25690891519393, 11.99348552764213, 12.17390135682884, 12.68575949533869, 13.26812947645242, 13.96877540787324, 14.18507385084618
