| L(s) = 1 | + 2·3-s + 5-s + 9-s − 11-s + 3·13-s + 2·15-s + 2·17-s + 5·19-s + 7·23-s + 25-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 + 45-s + 9·47-s + 4·51-s + 11·53-s − 55-s + 10·57-s − 8·59-s + 12·61-s + 3·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.516·15-s + 0.485·17-s + 1.14·19-s + 1.45·23-s + 1/5·25-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 + 0.149·45-s + 1.31·47-s + 0.560·51-s + 1.51·53-s − 0.134·55-s + 1.32·57-s − 1.04·59-s + 1.53·61-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.998419241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.998419241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 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
−9.091732998138559844786811523090, −8.597255807018327695840385730332, −7.58664983209587469841750189347, −7.15183087845906613930192459977, −5.82239473250670380943876399125, −5.31661172074901202674013179149, −3.95097779424508233039643008599, −3.21701786226494141604504472434, −2.39676310674079664374662908188, −1.20655101631795718573335213936,
1.20655101631795718573335213936, 2.39676310674079664374662908188, 3.21701786226494141604504472434, 3.95097779424508233039643008599, 5.31661172074901202674013179149, 5.82239473250670380943876399125, 7.15183087845906613930192459977, 7.58664983209587469841750189347, 8.597255807018327695840385730332, 9.091732998138559844786811523090
