| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·6-s − 4·7-s + 8-s − 2·9-s + 11-s − 2·12-s − 3·13-s + 8·14-s − 5·16-s − 14·17-s + 4·18-s + 3·19-s + 8·21-s − 2·22-s − 8·23-s − 2·24-s + 6·26-s + 9·27-s − 4·28-s − 5·29-s − 31-s + 8·32-s − 2·33-s + 28·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 1.63·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.301·11-s − 0.577·12-s − 0.832·13-s + 2.13·14-s − 5/4·16-s − 3.39·17-s + 0.942·18-s + 0.688·19-s + 1.74·21-s − 0.426·22-s − 1.66·23-s − 0.408·24-s + 1.17·26-s + 1.73·27-s − 0.755·28-s − 0.928·29-s − 0.179·31-s + 1.41·32-s − 0.348·33-s + 4.80·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 7 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 4 T + 22 T^{2} + 55 T^{3} + 22 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - T + 29 T^{2} - 23 T^{3} + 29 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 3 T^{2} - 25 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 14 T + 112 T^{2} + 555 T^{3} + 112 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 8 T + 60 T^{2} + 243 T^{3} + 60 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 5 T + 91 T^{2} + 285 T^{3} + 91 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + T + 63 T^{2} + 115 T^{3} + 63 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 5 T + p T^{2} - 25 T^{3} + p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - T + p T^{2} + 73 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 5 T + 16 T^{2} - 113 T^{3} + 16 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 77 T^{2} + 535 T^{3} + 77 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 31 T + 462 T^{2} + 4191 T^{3} + 462 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 6 T + 137 T^{2} + 508 T^{3} + 137 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 173 T^{2} - 367 T^{3} + 173 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 1573 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 7 T + 212 T^{2} - 947 T^{3} + 212 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + T + 189 T^{2} + 199 T^{3} + 189 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 254 T^{2} + 31 p T^{3} + 254 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 3 T - 21 T^{2} + 629 T^{3} - 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 20 T + 318 T^{2} + 3435 T^{3} + 318 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 2353 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.984787517994184317650238768979, −9.929999269860691039043409127078, −9.637232800057499945414840805923, −9.500048108954011372724931053585, −9.068602741112254889682895874804, −8.840502704715681938073667789669, −8.646068230999945421534801244418, −8.279703725424478488547000187623, −7.961411202189357606296853261964, −7.43647634996632542907942242136, −7.25111902854516608247191385495, −6.75772282115978843115342061758, −6.41707755124205882635825126959, −6.37029850032284263600142052056, −6.11372120983863813058396987031, −5.70185668950312104610222386817, −5.05224124579275937198814020153, −4.83824677001158109884965668823, −4.54389801042070457717316903227, −4.12650126542128489161939162915, −3.40565565240683285383965765507, −3.24813187253743835959875272829, −2.47046698591722202765913608213, −2.18803024657704188639462744284, −1.57249429351157364265463623964, 0, 0, 0,
1.57249429351157364265463623964, 2.18803024657704188639462744284, 2.47046698591722202765913608213, 3.24813187253743835959875272829, 3.40565565240683285383965765507, 4.12650126542128489161939162915, 4.54389801042070457717316903227, 4.83824677001158109884965668823, 5.05224124579275937198814020153, 5.70185668950312104610222386817, 6.11372120983863813058396987031, 6.37029850032284263600142052056, 6.41707755124205882635825126959, 6.75772282115978843115342061758, 7.25111902854516608247191385495, 7.43647634996632542907942242136, 7.961411202189357606296853261964, 8.279703725424478488547000187623, 8.646068230999945421534801244418, 8.840502704715681938073667789669, 9.068602741112254889682895874804, 9.500048108954011372724931053585, 9.637232800057499945414840805923, 9.929999269860691039043409127078, 9.984787517994184317650238768979
