| L(s) = 1 | − 2·3-s + 9-s − 11-s − 3·13-s − 2·17-s + 5·19-s − 7·23-s + 4·27-s − 6·29-s − 4·31-s + 2·33-s + 5·37-s + 6·39-s + 5·41-s − 6·43-s − 9·47-s + 4·51-s − 11·53-s − 10·57-s − 8·59-s + 12·61-s + 4·67-s + 14·69-s − 4·71-s + 12·73-s + 14·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.301·11-s − 0.832·13-s − 0.485·17-s + 1.14·19-s − 1.45·23-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.348·33-s + 0.821·37-s + 0.960·39-s + 0.780·41-s − 0.914·43-s − 1.31·47-s + 0.560·51-s − 1.51·53-s − 1.32·57-s − 1.04·59-s + 1.53·61-s + 0.488·67-s + 1.68·69-s − 0.474·71-s + 1.40·73-s + 1.57·79-s − 1.22·81-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
| \(L(1)\) |
\(\approx\) |
\(0.5363735399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5363735399\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.77059997448416529534220816906, −6.79501219038785126817744670843, −6.31845477471395476772162562238, −5.47322136436306136237330739940, −5.17401220985552024990338821821, −4.35884245233476222754031557133, −3.49551886654073396389115897134, −2.53578877672879065092789178056, −1.61767625530617632433976049487, −0.36458234217798345620679882508,
0.36458234217798345620679882508, 1.61767625530617632433976049487, 2.53578877672879065092789178056, 3.49551886654073396389115897134, 4.35884245233476222754031557133, 5.17401220985552024990338821821, 5.47322136436306136237330739940, 6.31845477471395476772162562238, 6.79501219038785126817744670843, 7.77059997448416529534220816906
