| L(s) = 1 | + 1.41·2-s − 2.64·5-s + 7-s − 2.82·8-s − 3.74·10-s − 5.74·13-s + 1.41·14-s − 4.00·16-s + 5.47·17-s + 5.74·19-s + 3.87·23-s + 2.00·25-s − 8.11·26-s + 3.87·29-s + 5.48·31-s + 7.74·34-s − 2.64·35-s + 3.48·37-s + 8.11·38-s + 7.48·40-s − 5.65·41-s − 43-s + 5.48·46-s + 5.47·47-s + 49-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 1.18·5-s + 0.377·7-s − 0.999·8-s − 1.18·10-s − 1.59·13-s + 0.377·14-s − 1.00·16-s + 1.32·17-s + 1.31·19-s + 0.808·23-s + 0.400·25-s − 1.59·26-s + 0.719·29-s + 0.984·31-s + 1.32·34-s − 0.447·35-s + 0.572·37-s + 1.31·38-s + 1.18·40-s − 0.883·41-s − 0.152·43-s + 0.808·46-s + 0.798·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 13 | \( 1 + 5.74T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 - 3.48T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 5.47T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 8.30T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 - 1.41T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5.83T + 89T^{2} \) |
| 97 | \( 1 + 7.22T + 97T^{2} \) |
| 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
−7.57011161749605087307977108989, −6.89408848161615281528888403887, −5.90194548175669496671433893437, −5.12449460439855861930337251663, −4.74262442628845806878341727539, −4.07535672915146882861037143408, −3.09412214047482753285799214724, −2.85768337670701013958104659881, −1.19189014388242068210239139133, 0,
1.19189014388242068210239139133, 2.85768337670701013958104659881, 3.09412214047482753285799214724, 4.07535672915146882861037143408, 4.74262442628845806878341727539, 5.12449460439855861930337251663, 5.90194548175669496671433893437, 6.89408848161615281528888403887, 7.57011161749605087307977108989
