| L(s) = 1 | + 3·2-s + 6·4-s + 6·5-s + 3·7-s + 10·8-s + 18·10-s − 6·13-s + 9·14-s + 15·16-s + 6·17-s + 36·20-s − 6·23-s + 12·25-s − 18·26-s + 18·28-s + 6·29-s − 15·31-s + 21·32-s + 18·34-s + 18·35-s + 3·37-s + 60·40-s + 9·41-s + 9·43-s − 18·46-s + 9·47-s + 6·49-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 2.68·5-s + 1.13·7-s + 3.53·8-s + 5.69·10-s − 1.66·13-s + 2.40·14-s + 15/4·16-s + 1.45·17-s + 8.04·20-s − 1.25·23-s + 12/5·25-s − 3.53·26-s + 3.40·28-s + 1.11·29-s − 2.69·31-s + 3.71·32-s + 3.08·34-s + 3.04·35-s + 0.493·37-s + 9.48·40-s + 1.40·41-s + 1.37·43-s − 2.65·46-s + 1.31·47-s + 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(55.04211912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(55.04211912\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 61 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 15 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 12 T^{2} - 37 T^{3} + 12 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 6 T + 42 T^{2} + 155 T^{3} + 42 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 207 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 6 T - 121 T^{3} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 6 T + 96 T^{2} - 349 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 15 T + 165 T^{2} + 1039 T^{3} + 165 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 66 T^{2} - 239 T^{3} + 66 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 9 T + 129 T^{2} - 685 T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 99 T^{2} - 523 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 9 T + 129 T^{2} - 775 T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 27 T + 390 T^{2} - 3491 T^{3} + 390 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 18 T + 222 T^{2} - 2133 T^{3} + 222 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 165 T^{2} - 383 T^{3} + 165 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 12 T - 24 T^{2} - 1041 T^{3} - 24 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 21 T + 303 T^{2} - 2819 T^{3} + 303 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 15 T + 273 T^{2} - 2247 T^{3} + 273 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 15 T + 249 T^{2} + 2189 T^{3} + 249 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 3 T + 213 T^{2} - 549 T^{3} + 213 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 45 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1861 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.11769333217839168566585682161, −6.57053911579892699554370176804, −6.39541108456756797706873584534, −6.37966030818257647790430206112, −5.88314745157325307900506997540, −5.76814355139124554169361139434, −5.42174688451378639576895953342, −5.35948956045552713772284268156, −5.32844023700967230934182145900, −5.26335154740789604695180428914, −4.76532643250096422829869298773, −4.38557885211296306896540085694, −4.12787830824594974699794396317, −3.83909151720908378820089120183, −3.78906998298105620876322944903, −3.61266485248636364005884405726, −2.70925165733635913764325084759, −2.60769611221493560973979216801, −2.53760291815271887236231625999, −2.28696581471441334828108479977, −2.13023354474899667262305470817, −1.69602017877664285911382971487, −1.42916699282280002646744914352, −1.04769100707904121533942794342, −0.61506146446814114498493642335,
0.61506146446814114498493642335, 1.04769100707904121533942794342, 1.42916699282280002646744914352, 1.69602017877664285911382971487, 2.13023354474899667262305470817, 2.28696581471441334828108479977, 2.53760291815271887236231625999, 2.60769611221493560973979216801, 2.70925165733635913764325084759, 3.61266485248636364005884405726, 3.78906998298105620876322944903, 3.83909151720908378820089120183, 4.12787830824594974699794396317, 4.38557885211296306896540085694, 4.76532643250096422829869298773, 5.26335154740789604695180428914, 5.32844023700967230934182145900, 5.35948956045552713772284268156, 5.42174688451378639576895953342, 5.76814355139124554169361139434, 5.88314745157325307900506997540, 6.37966030818257647790430206112, 6.39541108456756797706873584534, 6.57053911579892699554370176804, 7.11769333217839168566585682161
