| L(s) = 1 | − 8-s − 3·11-s − 12·19-s + 6·23-s − 12·29-s + 6·31-s + 6·41-s + 6·43-s − 24·47-s − 24·59-s − 18·61-s − 7·64-s + 12·67-s − 12·71-s − 24·73-s + 12·79-s − 18·83-s + 3·88-s + 18·89-s − 24·97-s − 12·101-s + 12·107-s − 12·109-s + 6·113-s + 6·121-s − 2·125-s + 127-s + ⋯ |
| L(s) = 1 | − 0.353·8-s − 0.904·11-s − 2.75·19-s + 1.25·23-s − 2.22·29-s + 1.07·31-s + 0.937·41-s + 0.914·43-s − 3.50·47-s − 3.12·59-s − 2.30·61-s − 7/8·64-s + 1.46·67-s − 1.42·71-s − 2.80·73-s + 1.35·79-s − 1.97·83-s + 0.319·88-s + 1.90·89-s − 2.43·97-s − 1.19·101-s + 1.16·107-s − 1.14·109-s + 0.564·113-s + 6/11·121-s − 0.178·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + T^{3} + p^{3} T^{6} \) |
| 5 | $D_{6}$ | \( 1 + 2 T^{3} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 24 T^{2} - 2 T^{3} + 24 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 27 T^{2} + 8 T^{3} + 27 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 12 T + 84 T^{2} + 420 T^{3} + 84 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 57 T^{2} - 244 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 12 T + 120 T^{2} + 702 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 57 T^{2} - 404 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 36 T^{2} - 246 T^{3} + 36 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 51 T^{2} - 460 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 117 T^{2} - 468 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 24 T + 312 T^{2} + 2584 T^{3} + 312 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 111 T^{2} + 120 T^{3} + 111 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 24 T + 264 T^{2} + 2116 T^{3} + 264 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 228 T^{2} - 1604 T^{3} + 228 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 165 T^{2} + 1320 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 24 T + 396 T^{2} + 3898 T^{3} + 396 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 189 T^{2} - 1640 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 309 T^{2} + 3036 T^{3} + 309 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 183 T^{2} - 1308 T^{3} + 183 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 24 T + 435 T^{2} + 4664 T^{3} + 435 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.66443206941579421518092226335, −7.60016297162296456478996497000, −7.17024162920674312725068779644, −7.01274410044055943659875231003, −6.68344204850119163243784793006, −6.30955678536473927368267228141, −6.20374071753601252736034225176, −6.12083857648148381471934616399, −5.70883961689617437781690966305, −5.66595855350543456443050258933, −5.11430643124302074258383270910, −4.83510247178727026349017465189, −4.70325819566239254849441120204, −4.55990453074554203913150791641, −4.11077705057673687655577701124, −4.05348135486003288935586484516, −3.54893493682141001499897370970, −3.18705713145536501806986653068, −3.00787552768999804729073298237, −2.82739662371277195267318374258, −2.38147198097828023193021057127, −2.16200538372962757306320816267, −1.66244486548185166315553131493, −1.48142452930858493181701750356, −1.12717103163536844651935152104, 0, 0, 0,
1.12717103163536844651935152104, 1.48142452930858493181701750356, 1.66244486548185166315553131493, 2.16200538372962757306320816267, 2.38147198097828023193021057127, 2.82739662371277195267318374258, 3.00787552768999804729073298237, 3.18705713145536501806986653068, 3.54893493682141001499897370970, 4.05348135486003288935586484516, 4.11077705057673687655577701124, 4.55990453074554203913150791641, 4.70325819566239254849441120204, 4.83510247178727026349017465189, 5.11430643124302074258383270910, 5.66595855350543456443050258933, 5.70883961689617437781690966305, 6.12083857648148381471934616399, 6.20374071753601252736034225176, 6.30955678536473927368267228141, 6.68344204850119163243784793006, 7.01274410044055943659875231003, 7.17024162920674312725068779644, 7.60016297162296456478996497000, 7.66443206941579421518092226335
