| L(s) = 1 | − 3·3-s + 3·5-s − 2·7-s + 6·9-s − 8·11-s + 2·13-s − 9·15-s − 4·17-s + 6·21-s + 3·23-s + 6·25-s − 10·27-s + 20·29-s − 14·31-s + 24·33-s − 6·35-s − 4·37-s − 6·39-s − 8·41-s + 8·43-s + 18·45-s − 12·47-s + 6·49-s + 12·51-s − 24·55-s − 14·59-s − 22·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s − 0.755·7-s + 2·9-s − 2.41·11-s + 0.554·13-s − 2.32·15-s − 0.970·17-s + 1.30·21-s + 0.625·23-s + 6/5·25-s − 1.92·27-s + 3.71·29-s − 2.51·31-s + 4.17·33-s − 1.01·35-s − 0.657·37-s − 0.960·39-s − 1.24·41-s + 1.21·43-s + 2.68·45-s − 1.75·47-s + 6/7·49-s + 1.68·51-s − 3.23·55-s − 1.82·59-s − 2.81·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T - 2 T^{2} - 16 T^{3} - 2 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 35 T^{2} + 112 T^{3} + 35 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} - 8 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 50 T^{2} + 134 T^{3} + 50 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 47 T^{2} - 8 T^{3} + 47 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 20 T + 214 T^{2} - 1410 T^{3} + 214 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 14 T + 86 T^{2} + 396 T^{3} + 86 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 92 T^{2} + 294 T^{3} + 92 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 8 T + 26 T^{2} - 14 T^{3} + 26 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 117 T^{2} - 608 T^{3} + 117 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 179 T^{2} + 1160 T^{3} + 179 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 110 T^{2} - 122 T^{3} + 110 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 108 T^{2} + 552 T^{3} + 108 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 22 T + 249 T^{2} + 2104 T^{3} + 249 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 26 T^{2} - 352 T^{3} + 26 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 80 T^{2} - 4 T^{3} + 80 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 89 T^{2} + 696 T^{3} + 89 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 141 T^{2} + 760 T^{3} + 141 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 22 T + 404 T^{2} + 4004 T^{3} + 404 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 179 T^{2} + 932 T^{3} + 179 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2 T + 43 T^{2} - 404 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.59379925235903249898669233788, −7.21142313550157766888958657005, −6.96735843547707639723555199673, −6.74686197456505457131150077996, −6.46621790040605593729036306032, −6.37888235372392597281462585962, −6.19747148837246034280457361880, −5.70983806368819028613914951523, −5.68019837925885163838617210115, −5.55699781568931905628552777582, −5.10492869083676477774891379815, −4.93530801048408249226542803945, −4.87837771860576898463518947924, −4.48435171572504983737314192155, −4.21205268443420562031152096122, −4.08152357908983427590955622803, −3.22222279573673447142068945571, −3.21810720804624491943046183880, −3.05280382271680339413591587885, −2.65124403697881943579303589311, −2.35964621995168022875338465977, −2.04233665602397301838800066253, −1.53949375176441335385279691145, −1.25584017134138386024805223605, −1.10524444463300926649713717407, 0, 0, 0,
1.10524444463300926649713717407, 1.25584017134138386024805223605, 1.53949375176441335385279691145, 2.04233665602397301838800066253, 2.35964621995168022875338465977, 2.65124403697881943579303589311, 3.05280382271680339413591587885, 3.21810720804624491943046183880, 3.22222279573673447142068945571, 4.08152357908983427590955622803, 4.21205268443420562031152096122, 4.48435171572504983737314192155, 4.87837771860576898463518947924, 4.93530801048408249226542803945, 5.10492869083676477774891379815, 5.55699781568931905628552777582, 5.68019837925885163838617210115, 5.70983806368819028613914951523, 6.19747148837246034280457361880, 6.37888235372392597281462585962, 6.46621790040605593729036306032, 6.74686197456505457131150077996, 6.96735843547707639723555199673, 7.21142313550157766888958657005, 7.59379925235903249898669233788
