| L(s) = 1 | + 3·2-s + 6·4-s + 6·5-s − 3·7-s + 10·8-s + 18·10-s + 12·11-s − 9·14-s + 15·16-s + 12·17-s + 36·20-s + 36·22-s + 12·23-s + 12·25-s − 18·28-s − 6·29-s + 3·31-s + 21·32-s + 36·34-s − 18·35-s − 3·37-s + 60·40-s − 9·41-s − 3·43-s + 72·44-s + 36·46-s + 15·47-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 + 3.61·11-s − 2.40·14-s + 15/4·16-s + 2.91·17-s + 8.04·20-s + 7.67·22-s + 2.50·23-s + 12/5·25-s − 3.40·28-s − 1.11·29-s + 0.538·31-s + 3.71·32-s + 6.17·34-s − 3.04·35-s − 0.493·37-s + 9.48·40-s − 1.40·41-s − 0.457·43-s + 10.8·44-s + 5.30·46-s + 2.18·47-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
| \(L(1)\) |
\(\approx\) |
\(74.03365698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(74.03365698\) |
| \(L(\frac{3}{2})\) |
|
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 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 63 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 43 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 12 T + 78 T^{2} - 315 T^{3} + 78 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_6$ | \( 1 - 19 T^{3} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 27 p T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 549 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 297 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 87 T^{2} - 169 T^{3} + 87 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 102 T^{2} + 203 T^{3} + 102 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 9 T + 3 p T^{2} + 711 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 259 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 15 T + 177 T^{2} - 1251 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 3 T + 78 T^{2} + 99 T^{3} + 78 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 60 T^{2} + 153 T^{3} + 60 p T^{4} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 3 T + 159 T^{2} + 367 T^{3} + 159 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 174 T^{2} - 715 T^{3} + 174 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 3 p T^{2} + 1197 T^{3} + 3 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 3 T - 15 T^{2} + 301 T^{3} - 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 9 T + 189 T^{2} + 1349 T^{3} + 189 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 3 T - 21 T^{2} - 1395 T^{3} - 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 3 T - 3 T^{2} + 1359 T^{3} - 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1051 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.86381111845563151266192525686, −6.59391781064700759509647144193, −6.48488162446852756865546962942, −6.40032665052189574051497640542, −6.05674550652505861447457829558, −5.83802776896166983517415191600, −5.64704653108366398115178661069, −5.35784637712948465808897734787, −5.33282714306288775315773455219, −5.19177310173975572808858940550, −4.61687633458230842254207904549, −4.29222556198303709428592248210, −4.23808850034289517442595163058, −3.72875184047482204218065270745, −3.57847557767442936949691747653, −3.49356617298099672000088178568, −2.96631804035046157320014324055, −2.92321599586458674492747993778, −2.81764246259747334122041258119, −2.05961133426776800664282421610, −1.76914424543634750095115935828, −1.72992300539698467345210638867, −1.28528908376966984438506609233, −1.17682675231216302799958834165, −0.75677045282674372361096562617,
0.75677045282674372361096562617, 1.17682675231216302799958834165, 1.28528908376966984438506609233, 1.72992300539698467345210638867, 1.76914424543634750095115935828, 2.05961133426776800664282421610, 2.81764246259747334122041258119, 2.92321599586458674492747993778, 2.96631804035046157320014324055, 3.49356617298099672000088178568, 3.57847557767442936949691747653, 3.72875184047482204218065270745, 4.23808850034289517442595163058, 4.29222556198303709428592248210, 4.61687633458230842254207904549, 5.19177310173975572808858940550, 5.33282714306288775315773455219, 5.35784637712948465808897734787, 5.64704653108366398115178661069, 5.83802776896166983517415191600, 6.05674550652505861447457829558, 6.40032665052189574051497640542, 6.48488162446852756865546962942, 6.59391781064700759509647144193, 6.86381111845563151266192525686
