| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s − 7-s + 8-s − 10-s − 2·13-s + 14-s − 15-s − 16-s + 17-s − 2·19-s + 21-s − 23-s − 24-s + 25-s + 2·26-s + 27-s + 30-s − 34-s − 35-s + 2·38-s + 2·39-s + 40-s − 42-s + 4·43-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s − 7-s + 8-s − 10-s − 2·13-s + 14-s − 15-s − 16-s + 17-s − 2·19-s + 21-s − 23-s − 24-s + 25-s + 2·26-s + 27-s + 30-s − 34-s − 35-s + 2·38-s + 2·39-s + 40-s − 42-s + 4·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2534134232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2534134232\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 23 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.485052248687138059902686915193, −9.422208367657054227276404566681, −8.874559361031550241763732835310, −8.581578548885687318789942771549, −8.069631815441039807909342175760, −7.52067856593741884769574865054, −7.27030387401504718161325695640, −6.93646818654059333188583174152, −6.29977604402308163691760224940, −6.05587643509983616092556166746, −5.66687729605093798875518242807, −5.37345287612802631110398413252, −4.71902444182632972769603304866, −4.27156610598315584945456650102, −4.09919578755265626987860645143, −3.01894111833612304237661069258, −2.52688947208610909347098015476, −2.25236102067172846031442935679, −1.39416191353706098976716249747, −0.47563871835096365482503940600,
0.47563871835096365482503940600, 1.39416191353706098976716249747, 2.25236102067172846031442935679, 2.52688947208610909347098015476, 3.01894111833612304237661069258, 4.09919578755265626987860645143, 4.27156610598315584945456650102, 4.71902444182632972769603304866, 5.37345287612802631110398413252, 5.66687729605093798875518242807, 6.05587643509983616092556166746, 6.29977604402308163691760224940, 6.93646818654059333188583174152, 7.27030387401504718161325695640, 7.52067856593741884769574865054, 8.069631815441039807909342175760, 8.581578548885687318789942771549, 8.874559361031550241763732835310, 9.422208367657054227276404566681, 9.485052248687138059902686915193
