| L(s) = 1 | − 3·3-s − 6·4-s + 6·9-s + 18·12-s − 3·13-s + 24·16-s − 6·17-s − 9·19-s + 6·23-s − 3·25-s − 10·27-s + 3·31-s − 36·36-s − 3·37-s + 9·39-s − 3·41-s − 21·43-s − 72·48-s + 18·51-s + 18·52-s − 6·53-s + 27·57-s + 18·59-s + 6·61-s − 80·64-s + 3·67-s + 36·68-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 3·4-s + 2·9-s + 5.19·12-s − 0.832·13-s + 6·16-s − 1.45·17-s − 2.06·19-s + 1.25·23-s − 3/5·25-s − 1.92·27-s + 0.538·31-s − 6·36-s − 0.493·37-s + 1.44·39-s − 0.468·41-s − 3.20·43-s − 10.3·48-s + 2.52·51-s + 2.49·52-s − 0.824·53-s + 3.57·57-s + 2.34·59-s + 0.768·61-s − 10·64-s + 0.366·67-s + 4.36·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T^{2} + 14 T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T^{2} - 50 T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 24 T^{2} + 43 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 T + 72 T^{2} + 347 T^{3} + 72 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 274 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 39 T^{2} - 112 T^{3} + 39 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 24 T^{2} - 79 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 + 21 T + 264 T^{2} + 2069 T^{3} + 264 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 39 T^{2} - 274 T^{3} + 39 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 99 T^{2} + 356 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 18 T + 213 T^{2} - 1764 T^{3} + 213 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 75 T^{2} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 3 T + 186 T^{2} - 403 T^{3} + 186 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 141 T^{2} + 144 T^{3} + 141 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 21 T + 354 T^{2} + 3329 T^{3} + 354 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 21 T + 312 T^{2} - 3391 T^{3} + 312 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 81 T^{2} + 194 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 12 T + 213 T^{2} - 1518 T^{3} + 213 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 207 T^{2} - 1660 T^{3} + 207 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.68837019280963736233619933560, −7.06550855759204635431204580473, −6.95061675969266775785980414949, −6.74742900582420963699454177155, −6.46529991545462644308780182470, −6.32432537618723417244912683554, −5.98116020008955187499438427970, −5.72082382544437866921316380236, −5.46967204088967709727643295713, −5.15442120296888724719273285726, −5.00389579240492155434169859711, −4.85726982539894124169091456327, −4.72848237468280045820264782656, −4.39082106293825466993555342403, −4.18523586899205603851906328021, −4.08074462883610043569560939193, −3.49955473946925690994148441940, −3.39808979162830408444577257341, −3.34680406097993601894936783108, −2.53164621767105562021420052121, −2.20135711510671317084357618805, −2.00650152791603545898143713172, −1.35082773287474788226758924938, −1.16326780040488763507288567066, −0.70958411150208033975693052155, 0, 0, 0,
0.70958411150208033975693052155, 1.16326780040488763507288567066, 1.35082773287474788226758924938, 2.00650152791603545898143713172, 2.20135711510671317084357618805, 2.53164621767105562021420052121, 3.34680406097993601894936783108, 3.39808979162830408444577257341, 3.49955473946925690994148441940, 4.08074462883610043569560939193, 4.18523586899205603851906328021, 4.39082106293825466993555342403, 4.72848237468280045820264782656, 4.85726982539894124169091456327, 5.00389579240492155434169859711, 5.15442120296888724719273285726, 5.46967204088967709727643295713, 5.72082382544437866921316380236, 5.98116020008955187499438427970, 6.32432537618723417244912683554, 6.46529991545462644308780182470, 6.74742900582420963699454177155, 6.95061675969266775785980414949, 7.06550855759204635431204580473, 7.68837019280963736233619933560
