| L(s) = 1 | + 2-s − 2.37·3-s + 4-s − 2.37·6-s + 2.37·7-s + 8-s + 2.62·9-s − 11-s − 2.37·12-s − 2·13-s + 2.37·14-s + 16-s + 4.37·17-s + 2.62·18-s + 6.37·19-s − 5.62·21-s − 22-s + 8.74·23-s − 2.37·24-s − 2·26-s + 0.883·27-s + 2.37·28-s − 4.37·29-s − 2.37·31-s + 32-s + 2.37·33-s + 4.37·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.36·3-s + 0.5·4-s − 0.968·6-s + 0.896·7-s + 0.353·8-s + 0.875·9-s − 0.301·11-s − 0.684·12-s − 0.554·13-s + 0.634·14-s + 0.250·16-s + 1.06·17-s + 0.619·18-s + 1.46·19-s − 1.22·21-s − 0.213·22-s + 1.82·23-s − 0.484·24-s − 0.392·26-s + 0.169·27-s + 0.448·28-s − 0.811·29-s − 0.426·31-s + 0.176·32-s + 0.412·33-s + 0.749·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.591240998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591240998\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 - 8.74T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 + 3.62T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 + 7.48T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
| 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
−11.12298809422988688774936192840, −10.30272152109640624176694306637, −9.177652837705956206745975582797, −7.65167153394625185273840922351, −7.14951069928516896711840529884, −5.74175392721250171252972121763, −5.33568156892610511001080614187, −4.52133639789801488398609656954, −3.00629857407444940403063989153, −1.19801306474962245344684043430,
1.19801306474962245344684043430, 3.00629857407444940403063989153, 4.52133639789801488398609656954, 5.33568156892610511001080614187, 5.74175392721250171252972121763, 7.14951069928516896711840529884, 7.65167153394625185273840922351, 9.177652837705956206745975582797, 10.30272152109640624176694306637, 11.12298809422988688774936192840
