| L(s) = 1 | + 5-s − 11-s + 2·13-s + 2·17-s − 4·19-s + 2·23-s + 25-s + 6·29-s + 2·31-s + 10·37-s + 8·41-s − 4·43-s + 4·47-s − 2·53-s − 55-s − 12·59-s − 10·61-s + 2·65-s + 4·67-s + 8·71-s − 14·73-s + 2·79-s − 2·83-s + 2·85-s + 10·89-s − 4·95-s + 8·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.417·23-s + 1/5·25-s + 1.11·29-s + 0.359·31-s + 1.64·37-s + 1.24·41-s − 0.609·43-s + 0.583·47-s − 0.274·53-s − 0.134·55-s − 1.56·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.949·71-s − 1.63·73-s + 0.225·79-s − 0.219·83-s + 0.216·85-s + 1.05·89-s − 0.410·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.210491501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.210491501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.59131194402541, −11.98696775320745, −11.60921512705557, −10.92682006203228, −10.71424623106401, −10.28093883774337, −9.754683726014479, −9.236431570728068, −8.943710182945093, −8.332236773104941, −7.874452706255302, −7.581801163472986, −6.822222661731436, −6.367824102617792, −6.044560765558245, −5.599731389031810, −4.879468198504047, −4.525710227395804, −4.044892434469266, −3.320570243869723, −2.793225142608516, −2.427224715920200, −1.648877052422394, −1.109390472617125, −0.4928344749444892,
0.4928344749444892, 1.109390472617125, 1.648877052422394, 2.427224715920200, 2.793225142608516, 3.320570243869723, 4.044892434469266, 4.525710227395804, 4.879468198504047, 5.599731389031810, 6.044560765558245, 6.367824102617792, 6.822222661731436, 7.581801163472986, 7.874452706255302, 8.332236773104941, 8.943710182945093, 9.236431570728068, 9.754683726014479, 10.28093883774337, 10.71424623106401, 10.92682006203228, 11.60921512705557, 11.98696775320745, 12.59131194402541
