| L(s) = 1 | + 3·3-s + 5-s − 9·7-s + 6·9-s + 2·11-s − 6·13-s + 3·15-s + 3·17-s + 2·19-s − 27·21-s − 23-s − 5·25-s + 10·27-s − 6·29-s − 5·31-s + 6·33-s − 9·35-s − 6·37-s − 18·39-s − 3·41-s + 14·43-s + 6·45-s + 8·47-s + 33·49-s + 9·51-s − 18·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 3.40·7-s + 2·9-s + 0.603·11-s − 1.66·13-s + 0.774·15-s + 0.727·17-s + 0.458·19-s − 5.89·21-s − 0.208·23-s − 25-s + 1.92·27-s − 1.11·29-s − 0.898·31-s + 1.04·33-s − 1.52·35-s − 0.986·37-s − 2.88·39-s − 0.468·41-s + 2.13·43-s + 0.894·45-s + 1.16·47-s + 33/7·49-s + 1.26·51-s − 2.47·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 251^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 251^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 251 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - T + 6 T^{2} + 3 T^{3} + 6 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} - 36 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 35 T^{2} + 148 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 26 T^{2} - 23 T^{3} + 26 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 49 T^{2} - 80 T^{3} + 49 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 60 T^{2} + 33 T^{3} + 60 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 59 T^{2} + 200 T^{3} + 59 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 80 T^{2} + 285 T^{3} + 80 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 83 T^{2} + 448 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 3 T + 78 T^{2} + 179 T^{3} + 78 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 161 T^{2} - 1224 T^{3} + 161 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 109 T^{2} - 480 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 18 T + 239 T^{2} + 1904 T^{3} + 239 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 18 T + 257 T^{2} + 2224 T^{3} + 257 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 155 T^{2} + 52 T^{3} + 155 p T^{4} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 5 T - 28 T^{2} + 1107 T^{3} - 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 34 T + 589 T^{2} + 6168 T^{3} + 589 p T^{4} + 34 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 23 T + 262 T^{2} + 2195 T^{3} + 262 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + T + 224 T^{2} + 153 T^{3} + 224 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 3 T + 224 T^{2} + 523 T^{3} + 224 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 135 T^{2} + 1220 T^{3} + 135 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 99 T^{2} - 496 T^{3} + 99 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.54933395841223706737722453865, −7.30905029763377229170502052654, −7.13936528435144306440339782475, −7.03490470176177087601293719771, −6.47292343352082859498809475768, −6.20191589359319221697397658969, −6.13899478617991310067054971727, −5.98692400305694581564979933435, −5.75603007903820395852316871484, −5.50159872686350911071707979309, −4.84904978918680255202697521638, −4.75674722447923001116116733075, −4.65965882107925816886746698211, −3.96908842226682820270011670502, −3.93092161204401181361621496586, −3.71254368307354495933070692492, −3.31644898079660977457719862105, −3.25368249435778958234583271249, −2.97928613900293090076003790074, −2.71968014207315359641223656945, −2.46605379940092345710339279509, −2.26734792171745550166660010790, −1.58415690006203426646410814218, −1.55345384756268912168041325654, −1.15323761316937351647333790951, 0, 0, 0,
1.15323761316937351647333790951, 1.55345384756268912168041325654, 1.58415690006203426646410814218, 2.26734792171745550166660010790, 2.46605379940092345710339279509, 2.71968014207315359641223656945, 2.97928613900293090076003790074, 3.25368249435778958234583271249, 3.31644898079660977457719862105, 3.71254368307354495933070692492, 3.93092161204401181361621496586, 3.96908842226682820270011670502, 4.65965882107925816886746698211, 4.75674722447923001116116733075, 4.84904978918680255202697521638, 5.50159872686350911071707979309, 5.75603007903820395852316871484, 5.98692400305694581564979933435, 6.13899478617991310067054971727, 6.20191589359319221697397658969, 6.47292343352082859498809475768, 7.03490470176177087601293719771, 7.13936528435144306440339782475, 7.30905029763377229170502052654, 7.54933395841223706737722453865
