| L(s) = 1 | + 2·3-s + 9-s + 4·11-s + 2·13-s − 2·19-s + 4·23-s − 4·27-s + 10·29-s − 4·31-s + 8·33-s + 2·37-s + 4·39-s + 12·41-s + 4·43-s + 4·47-s − 2·53-s − 4·57-s − 10·59-s − 6·61-s − 4·67-s + 8·69-s − 12·71-s − 4·73-s − 4·79-s − 11·81-s + 14·83-s + 20·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.458·19-s + 0.834·23-s − 0.769·27-s + 1.85·29-s − 0.718·31-s + 1.39·33-s + 0.328·37-s + 0.640·39-s + 1.87·41-s + 0.609·43-s + 0.583·47-s − 0.274·53-s − 0.529·57-s − 1.30·59-s − 0.768·61-s − 0.488·67-s + 0.963·69-s − 1.42·71-s − 0.468·73-s − 0.450·79-s − 1.22·81-s + 1.53·83-s + 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.769550862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.769550862\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.68507030579144555456871783500, −7.16759432126378501683775373383, −6.26293288182270398642489659736, −5.86356524950706943813938780822, −4.59392123946727139784327713209, −4.15968055360859092233618755516, −3.27446073558525328443554590463, −2.75990030136531674036409357976, −1.79537303139668617311032182765, −0.911333281697791217370263673115,
0.911333281697791217370263673115, 1.79537303139668617311032182765, 2.75990030136531674036409357976, 3.27446073558525328443554590463, 4.15968055360859092233618755516, 4.59392123946727139784327713209, 5.86356524950706943813938780822, 6.26293288182270398642489659736, 7.16759432126378501683775373383, 7.68507030579144555456871783500
