L(s) = 1 | − 8-s + 3·11-s − 2·27-s − 3·49-s + 3·83-s − 3·88-s − 6·107-s − 6·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 2·216-s + ⋯ |
L(s) = 1 | − 8-s + 3·11-s − 2·27-s − 3·49-s + 3·83-s − 3·88-s − 6·107-s − 6·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 2·216-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526905941\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526905941\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T^{3} + T^{6} \) |
| 19 | \( 1 \) |
good | 3 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 5 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 7 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 11 | \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \) |
| 13 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 17 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 23 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 29 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 31 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 37 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 41 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 43 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 47 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 53 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 59 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 61 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 67 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 71 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 73 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 79 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 83 | \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \) |
| 89 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 97 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.68709766483578484052480593346, −4.65551833989971582584397563549, −4.58378914598783496287340410022, −4.14757706686331138797609810281, −4.14482564202360871784907291852, −3.93525383357316514256716777302, −3.82228519154998574299724434764, −3.75739115220094840020928697475, −3.56087509032473436438334635633, −3.46728412778949314313551068734, −3.42724340133725705905199335827, −3.16788614294526589651645975774, −2.82071232895791343736159884846, −2.80295473625207870785760040799, −2.58776253955292085717042848389, −2.49061118893622838472850498626, −2.36719589324225337726965757802, −1.95828242719723953100016845398, −1.69878754120843026987059563009, −1.67243791021429230629917055859, −1.46458321187513046661434171785, −1.34219073662482448052155765958, −1.31871239397045894815729063763, −0.60910059503852922791290117459, −0.51616929634515816952066625186,
0.51616929634515816952066625186, 0.60910059503852922791290117459, 1.31871239397045894815729063763, 1.34219073662482448052155765958, 1.46458321187513046661434171785, 1.67243791021429230629917055859, 1.69878754120843026987059563009, 1.95828242719723953100016845398, 2.36719589324225337726965757802, 2.49061118893622838472850498626, 2.58776253955292085717042848389, 2.80295473625207870785760040799, 2.82071232895791343736159884846, 3.16788614294526589651645975774, 3.42724340133725705905199335827, 3.46728412778949314313551068734, 3.56087509032473436438334635633, 3.75739115220094840020928697475, 3.82228519154998574299724434764, 3.93525383357316514256716777302, 4.14482564202360871784907291852, 4.14757706686331138797609810281, 4.58378914598783496287340410022, 4.65551833989971582584397563549, 4.68709766483578484052480593346
Plot not available for L-functions of degree greater than 10.