| L(s) = 1 | − 3-s + 9-s + 2·11-s + 13-s − 2·17-s − 8·19-s + 4·23-s − 27-s − 6·29-s + 4·31-s − 2·33-s − 6·37-s − 39-s − 12·41-s + 4·43-s − 6·47-s − 7·49-s + 2·51-s + 2·53-s + 8·57-s + 14·59-s + 10·61-s − 4·67-s − 4·69-s − 2·71-s + 2·73-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s − 1.83·19-s + 0.834·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s − 0.160·39-s − 1.87·41-s + 0.609·43-s − 0.875·47-s − 49-s + 0.280·51-s + 0.274·53-s + 1.05·57-s + 1.82·59-s + 1.28·61-s − 0.488·67-s − 0.481·69-s − 0.237·71-s + 0.234·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.273316533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273316533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.08994555370664, −15.36061964426169, −14.95639660651947, −14.50301785820641, −13.63168730008551, −13.12773724866038, −12.74594279835974, −11.99819166311556, −11.47143735098502, −11.00147538849229, −10.40423957055678, −9.876774564903905, −9.046888169431588, −8.614978591773492, −8.020034402722511, −7.008577080453324, −6.667534842391160, −6.155508633545273, −5.280873586707268, −4.755563963620383, −3.970991970593315, −3.421656480811233, −2.259234506723220, −1.626857887033388, −0.5034519563741383,
0.5034519563741383, 1.626857887033388, 2.259234506723220, 3.421656480811233, 3.970991970593315, 4.755563963620383, 5.280873586707268, 6.155508633545273, 6.667534842391160, 7.008577080453324, 8.020034402722511, 8.614978591773492, 9.046888169431588, 9.876774564903905, 10.40423957055678, 11.00147538849229, 11.47143735098502, 11.99819166311556, 12.74594279835974, 13.12773724866038, 13.63168730008551, 14.50301785820641, 14.95639660651947, 15.36061964426169, 16.08994555370664
