| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s + 9·6-s + 3·7-s − 10·8-s + 6·9-s − 11-s − 18·12-s − 3·13-s − 9·14-s + 15·16-s − 3·17-s − 18·18-s − 19-s − 9·21-s + 3·22-s + 8·23-s + 30·24-s − 8·25-s + 9·26-s − 10·27-s + 18·28-s + 5·29-s − 11·31-s − 21·32-s + 3·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 3.67·6-s + 1.13·7-s − 3.53·8-s + 2·9-s − 0.301·11-s − 5.19·12-s − 0.832·13-s − 2.40·14-s + 15/4·16-s − 0.727·17-s − 4.24·18-s − 0.229·19-s − 1.96·21-s + 0.639·22-s + 1.66·23-s + 6.12·24-s − 8/5·25-s + 1.76·26-s − 1.92·27-s + 3.40·28-s + 0.928·29-s − 1.97·31-s − 3.71·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7390522788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7390522788\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 8 T^{2} - 7 T^{3} + 8 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + T + 24 T^{2} + 21 T^{3} + 24 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + T + 55 T^{2} + 37 T^{3} + 55 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 8 T + 2 p T^{2} - 171 T^{3} + 2 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 5 T + 79 T^{2} - 291 T^{3} + 79 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 11 T + 89 T^{2} + 471 T^{3} + 89 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 7 T + 3 p T^{2} + 511 T^{3} + 3 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - T + 65 T^{2} - 95 T^{3} + 65 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + T + 85 T^{2} + 169 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 21 T + 225 T^{2} - 1687 T^{3} + 225 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 5 T + 95 T^{2} + 349 T^{3} + 95 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 18 T + 4 p T^{2} + 1997 T^{3} + 4 p^{2} T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 11 T + 137 T^{2} - 795 T^{3} + 137 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 9 T + 137 T^{2} + 673 T^{3} + 137 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 7 T + 143 T^{2} - 7 p T^{3} + 143 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 170 T^{2} + 91 T^{3} + 170 p T^{4} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 9 T + 243 T^{2} + 1379 T^{3} + 243 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 19 T + 353 T^{2} - 3323 T^{3} + 353 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 7 T + 211 T^{2} + 1155 T^{3} + 211 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - T + 191 T^{2} - 375 T^{3} + 191 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.01274139313434753454894433980, −6.51174826473029923199348554728, −6.46154159074843284486028199124, −6.30940639219299616504385991159, −5.83718039186899719780107099037, −5.72355807338991134627038359179, −5.65906257509029265942713630512, −5.07922996772878189807895896625, −5.04103903789489447214342008912, −5.02107625750607882087771830333, −4.47251853175029433223933260840, −4.34075445934104165494669212549, −4.06503122846104606831830121512, −3.64730137946482113248331401707, −3.33633682287048552307981412121, −3.20099954218762037357830184396, −2.50995334697177708901348376541, −2.37906643135086861562512884819, −2.30149653878610191468231602888, −1.66654893750757856335957695330, −1.53814510236747509313625026831, −1.53517742638697454885116992421, −0.72517959806329538994456230932, −0.49871736264633848623295675104, −0.42578460582817182795199738573,
0.42578460582817182795199738573, 0.49871736264633848623295675104, 0.72517959806329538994456230932, 1.53517742638697454885116992421, 1.53814510236747509313625026831, 1.66654893750757856335957695330, 2.30149653878610191468231602888, 2.37906643135086861562512884819, 2.50995334697177708901348376541, 3.20099954218762037357830184396, 3.33633682287048552307981412121, 3.64730137946482113248331401707, 4.06503122846104606831830121512, 4.34075445934104165494669212549, 4.47251853175029433223933260840, 5.02107625750607882087771830333, 5.04103903789489447214342008912, 5.07922996772878189807895896625, 5.65906257509029265942713630512, 5.72355807338991134627038359179, 5.83718039186899719780107099037, 6.30940639219299616504385991159, 6.46154159074843284486028199124, 6.51174826473029923199348554728, 7.01274139313434753454894433980
