L(s) = 1 | + 2-s + 1.14·3-s + 4-s − 4.24·5-s + 1.14·6-s + 7-s + 8-s − 1.68·9-s − 4.24·10-s − 5.02·11-s + 1.14·12-s + 6.40·13-s + 14-s − 4.86·15-s + 16-s − 4.68·17-s − 1.68·18-s − 7.95·19-s − 4.24·20-s + 1.14·21-s − 5.02·22-s − 2.44·23-s + 1.14·24-s + 13.0·25-s + 6.40·26-s − 5.37·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.662·3-s + 0.5·4-s − 1.89·5-s + 0.468·6-s + 0.377·7-s + 0.353·8-s − 0.561·9-s − 1.34·10-s − 1.51·11-s + 0.331·12-s + 1.77·13-s + 0.267·14-s − 1.25·15-s + 0.250·16-s − 1.13·17-s − 0.397·18-s − 1.82·19-s − 0.948·20-s + 0.250·21-s − 1.07·22-s − 0.510·23-s + 0.234·24-s + 2.60·25-s + 1.25·26-s − 1.03·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1246 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1246 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 89 | \( 1 + T \) |
good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 7.95T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 - 2.96T + 37T^{2} \) |
| 41 | \( 1 + 0.815T + 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 + 5.68T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 + 4.37T + 61T^{2} \) |
| 67 | \( 1 - 8.99T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 + 5.38T + 73T^{2} \) |
| 79 | \( 1 + 3.84T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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
−8.750908621436080186948224236963, −8.364652631910741485739183572470, −7.895332400339827900610227701560, −6.89353404410198236926921028691, −5.88730933617347310250954272332, −4.70344691547169603324618180155, −3.97640257442474338092568858375, −3.28071944649577740324690275191, −2.20075576538961205031699827034, 0,
2.20075576538961205031699827034, 3.28071944649577740324690275191, 3.97640257442474338092568858375, 4.70344691547169603324618180155, 5.88730933617347310250954272332, 6.89353404410198236926921028691, 7.895332400339827900610227701560, 8.364652631910741485739183572470, 8.750908621436080186948224236963