| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 4·11-s − 4·13-s + 16-s + 3·18-s − 4·22-s + 2·23-s + 6·25-s + 4·26-s − 32-s − 3·36-s − 8·37-s + 4·44-s − 2·46-s + 2·49-s − 6·50-s − 4·52-s + 24·59-s − 16·61-s + 64-s + 3·72-s + 12·73-s + 8·74-s + 9·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s − 1.10·13-s + 1/4·16-s + 0.707·18-s − 0.852·22-s + 0.417·23-s + 6/5·25-s + 0.784·26-s − 0.176·32-s − 1/2·36-s − 1.31·37-s + 0.603·44-s − 0.294·46-s + 2/7·49-s − 0.848·50-s − 0.554·52-s + 3.12·59-s − 2.04·61-s + 1/8·64-s + 0.353·72-s + 1.40·73-s + 0.929·74-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008733098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008733098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.129184083267107993457187313853, −8.971935452602447588346290189052, −8.382468096458557457042578955575, −7.934797090973996806072777552438, −7.33780908492387851558326348269, −6.69441913641732873591204403077, −6.66534938649924724791949745826, −5.84432417959727908710252168629, −5.20363792939207012426517080907, −4.85472928398317566931109907696, −3.91541130112892355023471797064, −3.32533933556374790726950819131, −2.63232737474552247848840602025, −1.91815073870342461472602834954, −0.75831066970804062736803199920,
0.75831066970804062736803199920, 1.91815073870342461472602834954, 2.63232737474552247848840602025, 3.32533933556374790726950819131, 3.91541130112892355023471797064, 4.85472928398317566931109907696, 5.20363792939207012426517080907, 5.84432417959727908710252168629, 6.66534938649924724791949745826, 6.69441913641732873591204403077, 7.33780908492387851558326348269, 7.934797090973996806072777552438, 8.382468096458557457042578955575, 8.971935452602447588346290189052, 9.129184083267107993457187313853
