L(s) = 1 | + 2-s + 3·7-s − 13-s + 3·14-s + 17-s − 19-s + 23-s + 3·25-s − 26-s + 34-s − 37-s − 38-s − 3·41-s − 43-s + 46-s + 47-s + 6·49-s + 3·50-s − 74-s − 3·82-s − 86-s + 89-s − 3·91-s + 94-s − 97-s + 6·98-s + 101-s + ⋯ |
L(s) = 1 | + 2-s + 3·7-s − 13-s + 3·14-s + 17-s − 19-s + 23-s + 3·25-s − 26-s + 34-s − 37-s − 38-s − 3·41-s − 43-s + 46-s + 47-s + 6·49-s + 3·50-s − 74-s − 3·82-s − 86-s + 89-s − 3·91-s + 94-s − 97-s + 6·98-s + 101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.722861775\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.722861775\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 41 | $C_1$ | \( ( 1 + T )^{3} \) |
good | 2 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 17 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 19 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.976857762952515399555409853895, −7.932871141518870236024695316004, −7.48567206600870532646318121517, −7.28914788626878081825581172147, −7.00445360579493329952128281593, −6.85861175945590103652738174124, −6.66009108189749755568573142615, −6.19050079647528689482827761777, −5.68550212017814137575769996683, −5.59944926380598088895231367307, −5.13319449503001339350485020187, −5.05962152005061891764338708651, −4.93474610212934861695038140877, −4.60565933895355423887421195527, −4.48618270099382737373914540532, −4.36997224808589748301786221014, −3.59931104148218534627880623277, −3.48442627967473670769823122692, −3.27640570177130211155257930653, −2.54489260032120497780155444424, −2.54189142277243340413242525366, −1.86169295457560911973671366546, −1.83975269323596061561585091536, −1.09267674343460805980299570314, −1.09225969864206342020923818810,
1.09225969864206342020923818810, 1.09267674343460805980299570314, 1.83975269323596061561585091536, 1.86169295457560911973671366546, 2.54189142277243340413242525366, 2.54489260032120497780155444424, 3.27640570177130211155257930653, 3.48442627967473670769823122692, 3.59931104148218534627880623277, 4.36997224808589748301786221014, 4.48618270099382737373914540532, 4.60565933895355423887421195527, 4.93474610212934861695038140877, 5.05962152005061891764338708651, 5.13319449503001339350485020187, 5.59944926380598088895231367307, 5.68550212017814137575769996683, 6.19050079647528689482827761777, 6.66009108189749755568573142615, 6.85861175945590103652738174124, 7.00445360579493329952128281593, 7.28914788626878081825581172147, 7.48567206600870532646318121517, 7.932871141518870236024695316004, 7.976857762952515399555409853895