L(s) = 1 | − 2·3-s − 7-s + 9-s + 2·11-s + 4·13-s + 6·17-s − 4·19-s + 2·21-s + 23-s + 4·27-s − 6·29-s − 4·31-s − 4·33-s − 8·37-s − 8·39-s + 6·41-s + 6·43-s + 8·47-s + 49-s − 12·51-s − 4·53-s + 8·57-s − 14·59-s − 2·61-s − 63-s + 6·67-s − 2·69-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s + 0.436·21-s + 0.208·23-s + 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.696·33-s − 1.31·37-s − 1.28·39-s + 0.937·41-s + 0.914·43-s + 1.16·47-s + 1/7·49-s − 1.68·51-s − 0.549·53-s + 1.05·57-s − 1.82·59-s − 0.256·61-s − 0.125·63-s + 0.733·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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
−12.77371833582366, −12.56241240058401, −12.12426398995103, −11.74989857614168, −11.03465236421745, −10.75986830449368, −10.66943873201147, −9.774393289101746, −9.450833147042287, −8.809259558023554, −8.580189689452504, −7.703595476404917, −7.436968412616570, −6.762309635528928, −6.266470530729559, −5.922126369531490, −5.581161596748290, −5.033845098716235, −4.371300260971866, −3.764638381602431, −3.473210514029643, −2.750969741959835, −1.907162014538952, −1.301919614007197, −0.7414348913386630, 0,
0.7414348913386630, 1.301919614007197, 1.907162014538952, 2.750969741959835, 3.473210514029643, 3.764638381602431, 4.371300260971866, 5.033845098716235, 5.581161596748290, 5.922126369531490, 6.266470530729559, 6.762309635528928, 7.436968412616570, 7.703595476404917, 8.580189689452504, 8.809259558023554, 9.450833147042287, 9.774393289101746, 10.66943873201147, 10.75986830449368, 11.03465236421745, 11.74989857614168, 12.12426398995103, 12.56241240058401, 12.77371833582366