| L(s) = 1 | + 3·3-s + 3·7-s + 6·9-s − 6·11-s − 3·13-s + 9·21-s − 3·25-s + 10·27-s − 12·29-s − 15·31-s − 18·33-s + 3·37-s − 9·39-s − 12·41-s + 9·43-s − 18·47-s − 3·49-s + 6·59-s − 3·61-s + 18·63-s + 3·67-s − 6·71-s − 9·73-s − 9·75-s − 18·77-s − 27·79-s + 15·81-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.13·7-s + 2·9-s − 1.80·11-s − 0.832·13-s + 1.96·21-s − 3/5·25-s + 1.92·27-s − 2.22·29-s − 2.69·31-s − 3.13·33-s + 0.493·37-s − 1.44·39-s − 1.87·41-s + 1.37·43-s − 2.62·47-s − 3/7·49-s + 0.781·59-s − 0.384·61-s + 2.26·63-s + 0.366·67-s − 0.712·71-s − 1.05·73-s − 1.03·75-s − 2.05·77-s − 3.03·79-s + 5/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 3 T^{2} + 8 T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 12 T^{2} - 39 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 9 T^{2} - 4 T^{3} + 9 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 17 | $C_6$ | \( 1 + 3 T^{2} + 64 T^{3} + 3 p T^{4} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 57 T^{2} + 8 T^{3} + 57 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 504 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 15 T + 132 T^{2} + 803 T^{3} + 132 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 66 T^{2} - 3 p T^{3} + 66 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 792 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 144 T^{2} - 757 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 53 | $A_4\times C_2$ | \( 1 + 75 T^{2} - 296 T^{3} + 75 p T^{4} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 81 T^{2} - 284 T^{3} + 81 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 3 T + 138 T^{2} + 383 T^{3} + 138 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 3 T + 168 T^{2} - 439 T^{3} + 168 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 6 T + 81 T^{2} - 4 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 9 T + 198 T^{2} + 1133 T^{3} + 198 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 27 T + 396 T^{2} + 3943 T^{3} + 396 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 12 T + 153 T^{2} + 2056 T^{3} + 153 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 18 T + 339 T^{2} + 3132 T^{3} + 339 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 6 T + 159 T^{2} + 308 T^{3} + 159 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.32111366365350074581422153174, −7.28250087385235430928195722919, −6.90821726701866463773001099346, −6.79078545380025192134497466910, −6.20847736599672451544253035399, −5.88373871021845457818800467961, −5.86073871698984008925083523120, −5.46718442670513106905095638965, −5.33398173846403903220510429581, −5.10388036251243543632860327330, −4.82220158988430698331576695780, −4.61346080245343535547457420056, −4.42712243405257729728345888730, −3.92397261493310377912322884568, −3.85783654910241728965712789503, −3.65657363216948497865891259454, −3.19188180954231695822227522960, −3.10175522928482241903127756479, −2.77964933989581173543385995107, −2.45958127768957480839501908250, −2.23291025188810801818398512316, −2.00201159338040773759171775047, −1.60909202142343944445916908517, −1.46128090213784122120554665086, −1.28478851212430623206596943854, 0, 0, 0,
1.28478851212430623206596943854, 1.46128090213784122120554665086, 1.60909202142343944445916908517, 2.00201159338040773759171775047, 2.23291025188810801818398512316, 2.45958127768957480839501908250, 2.77964933989581173543385995107, 3.10175522928482241903127756479, 3.19188180954231695822227522960, 3.65657363216948497865891259454, 3.85783654910241728965712789503, 3.92397261493310377912322884568, 4.42712243405257729728345888730, 4.61346080245343535547457420056, 4.82220158988430698331576695780, 5.10388036251243543632860327330, 5.33398173846403903220510429581, 5.46718442670513106905095638965, 5.86073871698984008925083523120, 5.88373871021845457818800467961, 6.20847736599672451544253035399, 6.79078545380025192134497466910, 6.90821726701866463773001099346, 7.28250087385235430928195722919, 7.32111366365350074581422153174
