| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s + 3·13-s − 14-s + 16-s + 8·17-s − 3·19-s − 20-s + 22-s − 6·23-s + 25-s + 3·26-s − 28-s + 6·29-s − 4·31-s + 32-s + 8·34-s + 35-s + 2·37-s − 3·38-s − 40-s + 11·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 1.94·17-s − 0.688·19-s − 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.588·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.37·34-s + 0.169·35-s + 0.328·37-s − 0.486·38-s − 0.158·40-s + 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.649932581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.649932581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.262376083734799386446326567048, −8.193558722618938481714595933668, −7.71612958689988930627770884554, −6.64050986012999694700244620035, −6.01556105348177316652884975092, −5.22198227102056816747420800181, −4.01475654289198334733173459253, −3.61495074489563682100896753660, −2.47177520409495365376892223744, −1.04434040928825976495251355636,
1.04434040928825976495251355636, 2.47177520409495365376892223744, 3.61495074489563682100896753660, 4.01475654289198334733173459253, 5.22198227102056816747420800181, 6.01556105348177316652884975092, 6.64050986012999694700244620035, 7.71612958689988930627770884554, 8.193558722618938481714595933668, 9.262376083734799386446326567048
