L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s − 6·5-s − 9·6-s + 3·7-s + 10·8-s + 6·9-s − 18·10-s − 18·12-s + 6·13-s + 9·14-s + 18·15-s + 15·16-s − 6·17-s + 18·18-s − 36·20-s − 9·21-s + 6·23-s − 30·24-s + 12·25-s + 18·26-s − 10·27-s + 18·28-s + 6·29-s + 54·30-s + 15·31-s + ⋯ |
L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 2.68·5-s − 3.67·6-s + 1.13·7-s + 3.53·8-s + 2·9-s − 5.69·10-s − 5.19·12-s + 1.66·13-s + 2.40·14-s + 4.64·15-s + 15/4·16-s − 1.45·17-s + 4.24·18-s − 8.04·20-s − 1.96·21-s + 1.25·23-s − 6.12·24-s + 12/5·25-s + 3.53·26-s − 1.92·27-s + 3.40·28-s + 1.11·29-s + 9.85·30-s + 2.69·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.201820125\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.201820125\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | | \( 1 \) |
good | 5 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 61 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 15 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 12 T^{2} + 37 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 155 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 121 T^{3} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 6 T + 96 T^{2} - 349 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 15 T + 165 T^{2} - 1039 T^{3} + 165 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 66 T^{2} + 239 T^{3} + 66 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 9 T + 129 T^{2} - 685 T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 99 T^{2} - 523 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 129 T^{2} + 775 T^{3} + 129 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 27 T + 390 T^{2} - 3491 T^{3} + 390 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 18 T + 222 T^{2} - 2133 T^{3} + 222 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 165 T^{2} - 383 T^{3} + 165 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 12 T - 24 T^{2} + 1041 T^{3} - 24 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 21 T + 303 T^{2} - 2819 T^{3} + 303 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 15 T + 273 T^{2} - 2247 T^{3} + 273 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 15 T + 249 T^{2} - 2189 T^{3} + 249 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 3 T + 213 T^{2} + 549 T^{3} + 213 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 45 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1861 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.007163940479088417142477848001, −7.56590854427988486384970129250, −7.51344458179378749337477746297, −6.97860466397803875160037265232, −6.86242043576251519762161699678, −6.72055175034900457034171013075, −6.51141979290713474130333236007, −5.97004086615526249419146877733, −5.95223525895668392270859585770, −5.61710732190131596883591199080, −5.18542897435060327636616578344, −4.87839641419007151221110633635, −4.74482983418777640433021976555, −4.63106896840390127342649969178, −4.15488237515851959146740096688, −4.05489412568529068171761198682, −3.76794019526435460596299440498, −3.48776306539571719315407114457, −3.41430015207935316398647035398, −2.42122008440997029676632606496, −2.38004122342987678282184697171, −2.13750674071357203653735324162, −1.01566411363640445936628896811, −0.839624117923517940111770228900, −0.804558554932805246678121583302,
0.804558554932805246678121583302, 0.839624117923517940111770228900, 1.01566411363640445936628896811, 2.13750674071357203653735324162, 2.38004122342987678282184697171, 2.42122008440997029676632606496, 3.41430015207935316398647035398, 3.48776306539571719315407114457, 3.76794019526435460596299440498, 4.05489412568529068171761198682, 4.15488237515851959146740096688, 4.63106896840390127342649969178, 4.74482983418777640433021976555, 4.87839641419007151221110633635, 5.18542897435060327636616578344, 5.61710732190131596883591199080, 5.95223525895668392270859585770, 5.97004086615526249419146877733, 6.51141979290713474130333236007, 6.72055175034900457034171013075, 6.86242043576251519762161699678, 6.97860466397803875160037265232, 7.51344458179378749337477746297, 7.56590854427988486384970129250, 8.007163940479088417142477848001