| L(s) = 1 | − 3·3-s − 3·5-s − 2·7-s + 6·9-s + 4·11-s − 6·13-s + 9·15-s + 3·19-s + 6·21-s − 8·23-s + 6·25-s − 10·27-s + 10·29-s − 6·31-s − 12·33-s + 6·35-s − 6·37-s + 18·39-s + 4·41-s − 18·45-s − 16·47-s + 49-s + 22·53-s − 12·55-s − 9·57-s + 22·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s + 1.20·11-s − 1.66·13-s + 2.32·15-s + 0.688·19-s + 1.30·21-s − 1.66·23-s + 6/5·25-s − 1.92·27-s + 1.85·29-s − 1.07·31-s − 2.08·33-s + 1.01·35-s − 0.986·37-s + 2.88·39-s + 0.624·41-s − 2.68·45-s − 2.33·47-s + 1/7·49-s + 3.02·53-s − 1.61·55-s − 1.19·57-s + 2.86·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9521423313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9521423313\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 3 T^{2} - 16 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 19 T^{2} - 96 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 152 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 11 T^{2} + 64 T^{3} + 11 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 65 T^{2} + 352 T^{3} + 65 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 69 T^{2} - 312 T^{3} + 69 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 65 T^{2} + 236 T^{3} + 65 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 56 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} + 72 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 39 T^{2} + 216 T^{3} + 39 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 16 T + 201 T^{2} + 1488 T^{3} + 201 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 22 T + 295 T^{2} - 2508 T^{3} + 295 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 22 T + 217 T^{2} - 1620 T^{3} + 217 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 87 T^{2} - 404 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 105 T^{2} - 664 T^{3} + 105 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 29 T^{2} - 464 T^{3} + 29 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 119 T^{2} - 972 T^{3} + 119 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 10 T + 3 p T^{2} + 1516 T^{3} + 3 p^{2} T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 149 T^{2} - 4 T^{3} + 149 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 24 T + 369 T^{2} - 3848 T^{3} + 369 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 213 T^{2} + 8 p T^{3} + 213 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.116027610111035396979074700631, −7.40529565216148883397780339766, −7.34330198014524915459169362120, −7.31282024172218316625661097880, −6.85820244991885569701212296829, −6.70184761449653474416459435094, −6.61728795118162019768568763515, −6.09810116047250525231431237307, −5.79738706471768526257391846027, −5.75002105424631127636689219311, −5.35292408680954214046335593835, −4.93041491810347371050434509854, −4.79551937360616566535910273697, −4.46336391010847641818491404612, −4.34549328540183803127442306463, −3.86595299171695745071788446635, −3.52798538739300512165184170061, −3.43772228170649254482369096433, −3.13419697385842829772938121604, −2.32292200278481330769654759971, −2.18558175551321423842422298761, −1.82304917472216000903910699258, −1.03282597749396725796124261119, −0.56228806486245403795995779142, −0.50219645841858026346765490631,
0.50219645841858026346765490631, 0.56228806486245403795995779142, 1.03282597749396725796124261119, 1.82304917472216000903910699258, 2.18558175551321423842422298761, 2.32292200278481330769654759971, 3.13419697385842829772938121604, 3.43772228170649254482369096433, 3.52798538739300512165184170061, 3.86595299171695745071788446635, 4.34549328540183803127442306463, 4.46336391010847641818491404612, 4.79551937360616566535910273697, 4.93041491810347371050434509854, 5.35292408680954214046335593835, 5.75002105424631127636689219311, 5.79738706471768526257391846027, 6.09810116047250525231431237307, 6.61728795118162019768568763515, 6.70184761449653474416459435094, 6.85820244991885569701212296829, 7.31282024172218316625661097880, 7.34330198014524915459169362120, 7.40529565216148883397780339766, 8.116027610111035396979074700631
