| L(s) = 1 | − 3·5-s − 3·7-s + 6·13-s − 6·19-s + 3·23-s + 6·25-s − 3·29-s − 6·31-s + 9·35-s + 12·37-s + 3·41-s − 6·43-s − 15·47-s + 3·49-s − 6·53-s − 6·59-s + 21·61-s − 18·65-s − 9·67-s − 24·71-s + 24·73-s − 6·79-s + 21·83-s − 9·89-s − 18·91-s + 18·95-s + 18·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s + 1.66·13-s − 1.37·19-s + 0.625·23-s + 6/5·25-s − 0.557·29-s − 1.07·31-s + 1.52·35-s + 1.97·37-s + 0.468·41-s − 0.914·43-s − 2.18·47-s + 3/7·49-s − 0.824·53-s − 0.781·59-s + 2.68·61-s − 2.23·65-s − 1.09·67-s − 2.84·71-s + 2.80·73-s − 0.675·79-s + 2.30·83-s − 0.953·89-s − 1.88·91-s + 1.84·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5286355235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5286355235\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} - T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 36 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 27 T^{2} - 80 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 27 T^{2} + 36 T^{3} + 27 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 224 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 3 T + 54 T^{2} - 147 T^{3} + 54 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 3 T + 18 T^{2} - 105 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 81 T^{2} + 368 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 75 T^{2} - 452 T^{3} + 75 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} - 327 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 117 T^{2} + 440 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 15 T + 180 T^{2} + 1437 T^{3} + 180 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 99 T^{2} + 708 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 117 T^{2} + 780 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 21 T + 246 T^{2} - 2153 T^{3} + 246 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 180 T^{2} + 1055 T^{3} + 180 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 24 T + 321 T^{2} + 3084 T^{3} + 321 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 21 T + 378 T^{2} - 3729 T^{3} + 378 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 327 T^{2} - 3068 T^{3} + 327 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.25411462819338033534820483885, −6.70085839154317251911500052782, −6.58687546665492817851526617533, −6.43538864250594464739634152578, −6.15088652312198148046112924374, −5.96451320503968049475869220434, −5.92685198770698075100940751812, −5.30624119746673392821113529692, −5.13082376138331137448603458269, −4.86853283002918350706362200271, −4.61926478012507911739449227423, −4.38129928481693233209396609603, −4.03583059767705342516496911756, −3.64469748449226429251815347311, −3.63050748928672314431079193280, −3.60192192736107681485561747968, −3.11768962505041152190296152963, −2.83780903318345519852329529605, −2.57067568434068252148916631910, −2.20020613252693399681801045462, −1.68156753290685221525260537872, −1.58646099421254838983079143706, −0.970210919749162807771065136279, −0.68161783792182381065065988132, −0.16055882980156606808317701033,
0.16055882980156606808317701033, 0.68161783792182381065065988132, 0.970210919749162807771065136279, 1.58646099421254838983079143706, 1.68156753290685221525260537872, 2.20020613252693399681801045462, 2.57067568434068252148916631910, 2.83780903318345519852329529605, 3.11768962505041152190296152963, 3.60192192736107681485561747968, 3.63050748928672314431079193280, 3.64469748449226429251815347311, 4.03583059767705342516496911756, 4.38129928481693233209396609603, 4.61926478012507911739449227423, 4.86853283002918350706362200271, 5.13082376138331137448603458269, 5.30624119746673392821113529692, 5.92685198770698075100940751812, 5.96451320503968049475869220434, 6.15088652312198148046112924374, 6.43538864250594464739634152578, 6.58687546665492817851526617533, 6.70085839154317251911500052782, 7.25411462819338033534820483885
