| L(s) = 1 | + 3·3-s − 2·7-s + 6·9-s + 4·11-s − 6·13-s + 4·17-s + 6·19-s − 6·21-s + 3·23-s + 25-s + 10·27-s + 2·29-s + 4·31-s + 12·33-s − 14·37-s − 18·39-s − 2·41-s − 2·43-s + 16·47-s − 5·49-s + 12·51-s + 4·53-s + 18·57-s − 12·59-s − 22·61-s − 12·63-s + 2·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.755·7-s + 2·9-s + 1.20·11-s − 1.66·13-s + 0.970·17-s + 1.37·19-s − 1.30·21-s + 0.625·23-s + 1/5·25-s + 1.92·27-s + 0.371·29-s + 0.718·31-s + 2.08·33-s − 2.30·37-s − 2.88·39-s − 0.312·41-s − 0.304·43-s + 2.33·47-s − 5/7·49-s + 1.68·51-s + 0.549·53-s + 2.38·57-s − 1.56·59-s − 2.81·61-s − 1.51·63-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.269975628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.269975628\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 20 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 43 T^{2} - 120 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 220 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 45 T^{2} - 184 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 14 T + 139 T^{2} + 884 T^{3} + 139 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 39 T^{2} + 60 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 77 T^{2} - 12 T^{3} + 77 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 16 T + 173 T^{2} - 1376 T^{3} + 173 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 15 T^{2} + 168 T^{3} + 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 161 T^{2} + 1352 T^{3} + 161 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 22 T + 307 T^{2} + 2884 T^{3} + 307 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 149 T^{2} - 84 T^{3} + 149 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 85 T^{2} - 1392 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 - 26 T + 449 T^{2} - 4644 T^{3} + 449 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 73 T^{2} + 504 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 227 T^{2} + 1120 T^{3} + 227 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 335 T^{2} - 3196 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.43949785788312569324987124121, −7.34392379727034108292487532656, −7.21119074316047271385014126926, −6.60183655505472391739538942184, −6.44367948805154453125882944983, −6.29666805689443437097105519520, −6.16735882831077594015077152835, −5.52193128333286933597122321135, −5.34909417191499042338381864691, −5.06766370451500861732498061167, −4.84520439998880821060633731084, −4.59113175451670621091919862042, −4.39111976349798637207706949802, −3.74795794869359279904455040671, −3.72930279342896227522798491816, −3.51671604833098938828319678058, −3.11854852339768771460512680952, −2.92700411661605780429272731554, −2.84996187048988641452208872814, −2.31962777461362347603453002252, −1.92332463200877109556400780449, −1.76900906091060646633401155993, −1.31660757416818293654552097499, −0.818388173607787936613950388616, −0.52693120517922505271255441437,
0.52693120517922505271255441437, 0.818388173607787936613950388616, 1.31660757416818293654552097499, 1.76900906091060646633401155993, 1.92332463200877109556400780449, 2.31962777461362347603453002252, 2.84996187048988641452208872814, 2.92700411661605780429272731554, 3.11854852339768771460512680952, 3.51671604833098938828319678058, 3.72930279342896227522798491816, 3.74795794869359279904455040671, 4.39111976349798637207706949802, 4.59113175451670621091919862042, 4.84520439998880821060633731084, 5.06766370451500861732498061167, 5.34909417191499042338381864691, 5.52193128333286933597122321135, 6.16735882831077594015077152835, 6.29666805689443437097105519520, 6.44367948805154453125882944983, 6.60183655505472391739538942184, 7.21119074316047271385014126926, 7.34392379727034108292487532656, 7.43949785788312569324987124121
