| L(s) = 1 | − 3·3-s − 3·5-s − 2·7-s + 6·9-s − 2·11-s − 6·13-s + 9·15-s + 4·17-s + 8·19-s + 6·21-s − 14·23-s + 6·25-s − 10·27-s + 16·29-s + 3·31-s + 6·33-s + 6·35-s − 8·37-s + 18·39-s + 4·41-s + 2·43-s − 18·45-s − 14·47-s − 5·49-s − 12·51-s + 8·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s − 0.603·11-s − 1.66·13-s + 2.32·15-s + 0.970·17-s + 1.83·19-s + 1.30·21-s − 2.91·23-s + 6/5·25-s − 1.92·27-s + 2.97·29-s + 0.538·31-s + 1.04·33-s + 1.01·35-s − 1.31·37-s + 2.88·39-s + 0.624·41-s + 0.304·43-s − 2.68·45-s − 2.04·47-s − 5/7·49-s − 1.68·51-s + 1.09·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 38 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 + 6 T + 47 T^{2} + 154 T^{3} + 47 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 116 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 272 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 14 T + 129 T^{2} + 720 T^{3} + 129 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 16 T + 149 T^{2} - 938 T^{3} + 149 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 14 p T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 35 T^{2} - 344 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 69 T^{2} + 28 T^{3} + 69 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 14 T + 49 T^{2} - 72 T^{3} + 49 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 87 T^{2} - 948 T^{3} + 87 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 26 T + 379 T^{2} - 3534 T^{3} + 379 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 18 T + 227 T^{2} + 1900 T^{3} + 227 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 65 T^{2} + 274 T^{3} + 65 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 215 T^{2} + 566 T^{3} + 215 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 12 T + 155 T^{2} + 1618 T^{3} + 155 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 93 T^{2} + 1132 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 205 T^{2} - 1272 T^{3} + 205 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 249 T^{2} + 1018 T^{3} + 249 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 51 T^{2} + 160 T^{3} + 51 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.33199746114560002404550694809, −7.02634354885657046724200185133, −6.81231560843725653787277020085, −6.75055937615609028870648443397, −6.25481871945585765187968458295, −6.20632792002260153690078783121, −6.03415055900949346547384566390, −5.54969747706520368649703898337, −5.42936145249297449834853540772, −5.26600149863978028534976888103, −4.87144279939083675148566421451, −4.84284934859947163901201135902, −4.56054640293132730132078593012, −4.23301872034711395075252557585, −3.99485206406728884356889805760, −3.84363679225316951164460429935, −3.26484678714390469878733802316, −3.18046875394932111684541247204, −3.07959408390127984422828810773, −2.56392287784121378460895300877, −2.15763720323436407413451511825, −2.12807715079537252657870700410, −1.20135660405723317324864591530, −1.18954553184534686353777954147, −0.888185723246622037030507730206, 0, 0, 0,
0.888185723246622037030507730206, 1.18954553184534686353777954147, 1.20135660405723317324864591530, 2.12807715079537252657870700410, 2.15763720323436407413451511825, 2.56392287784121378460895300877, 3.07959408390127984422828810773, 3.18046875394932111684541247204, 3.26484678714390469878733802316, 3.84363679225316951164460429935, 3.99485206406728884356889805760, 4.23301872034711395075252557585, 4.56054640293132730132078593012, 4.84284934859947163901201135902, 4.87144279939083675148566421451, 5.26600149863978028534976888103, 5.42936145249297449834853540772, 5.54969747706520368649703898337, 6.03415055900949346547384566390, 6.20632792002260153690078783121, 6.25481871945585765187968458295, 6.75055937615609028870648443397, 6.81231560843725653787277020085, 7.02634354885657046724200185133, 7.33199746114560002404550694809
