| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 9-s + 2·11-s − 13-s + 4·15-s + 17-s + 8·19-s − 4·21-s + 2·23-s − 25-s − 4·27-s + 6·29-s + 10·31-s + 4·33-s − 4·35-s + 2·37-s − 2·39-s − 10·41-s − 8·43-s + 2·45-s − 3·49-s + 2·51-s + 6·53-s + 4·55-s + 16·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 1.03·15-s + 0.242·17-s + 1.83·19-s − 0.872·21-s + 0.417·23-s − 1/5·25-s − 0.769·27-s + 1.11·29-s + 1.79·31-s + 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.320·39-s − 1.56·41-s − 1.21·43-s + 0.298·45-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 2.11·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.362386766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.362386766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.583574618989073679231801216525, −8.023857679001636458245746242222, −7.03791415844026979307541177555, −6.46535278223783141236409175290, −5.58806801006060408556500960666, −4.76968023857666672498802922998, −3.51375486629792750099552508314, −3.06212003927545198461594716885, −2.20726477985392000599426873757, −1.07914707654932377405295976693,
1.07914707654932377405295976693, 2.20726477985392000599426873757, 3.06212003927545198461594716885, 3.51375486629792750099552508314, 4.76968023857666672498802922998, 5.58806801006060408556500960666, 6.46535278223783141236409175290, 7.03791415844026979307541177555, 8.023857679001636458245746242222, 8.583574618989073679231801216525
