| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s − 9·6-s − 3·7-s + 10·8-s + 6·9-s + 11-s − 18·12-s − 9·14-s + 15·16-s + 9·17-s + 18·18-s + 19-s + 9·21-s + 3·22-s + 23-s − 30·24-s − 8·25-s − 10·27-s − 18·28-s − 2·29-s + 4·31-s + 21·32-s − 3·33-s + 27·34-s + 36·36-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s − 3.67·6-s − 1.13·7-s + 3.53·8-s + 2·9-s + 0.301·11-s − 5.19·12-s − 2.40·14-s + 15/4·16-s + 2.18·17-s + 4.24·18-s + 0.229·19-s + 1.96·21-s + 0.639·22-s + 0.208·23-s − 6.12·24-s − 8/5·25-s − 1.92·27-s − 3.40·28-s − 0.371·29-s + 0.718·31-s + 3.71·32-s − 0.522·33-s + 4.63·34-s + 6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.36478101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.36478101\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 8 T^{2} + 7 T^{3} + 8 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - T + 17 T^{2} + 7 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 9 T + 57 T^{2} - 277 T^{3} + 57 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - T + 41 T^{2} - 9 T^{3} + 41 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - T + 25 T^{2} - 129 T^{3} + 25 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 2 T + 2 p T^{2} + 45 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 4 T + 54 T^{2} - 79 T^{3} + 54 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 14 T + 146 T^{2} - 1029 T^{3} + 146 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 6 T + 86 T^{2} - 11 p T^{3} + 86 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 29 T + 407 T^{2} + 3375 T^{3} + 407 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 7 T + 120 T^{2} - 567 T^{3} + 120 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 16 T + 214 T^{2} - 1653 T^{3} + 214 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 10 T + 208 T^{2} + 1209 T^{3} + 208 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 11 T + 186 T^{2} - 1271 T^{3} + 186 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 2 T + 116 T^{2} - 519 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 14 T + 241 T^{2} - 1932 T^{3} + 241 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 7 T + 9 T^{2} + 245 T^{3} + 9 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 2 T + 26 T^{2} - 945 T^{3} + 26 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 5 T + 171 T^{2} + 411 T^{3} + 171 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 120 T^{2} - 497 T^{3} + 120 p T^{4} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 6 T + 86 T^{2} - 1921 T^{3} + 86 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.97571308322249683287413505120, −6.67737889969777734493315144302, −6.31114603329833797337362154568, −6.20834011992144226178335123324, −5.91868047272880425165128077794, −5.77927732917681737026692716288, −5.73214210635757272560599836299, −5.27288193137734556227974655274, −5.16264770486478922947440877943, −5.08897951785586896716068707039, −4.49454858810118512447501815079, −4.47625231756729024719877778905, −4.15662587948371313749634278917, −3.83757511907556622262080628487, −3.66938354277451455100568008582, −3.48617716682372907270574098783, −3.03652081732347157195545113311, −2.99267196603154114743720235527, −2.63412286386940625106858155014, −2.08178636012757952688851218408, −1.82253461326485157326253031006, −1.69405882352552328190478774719, −0.990368982624148812032734833212, −0.70419552267290399117817170350, −0.56881597555184533844271705534,
0.56881597555184533844271705534, 0.70419552267290399117817170350, 0.990368982624148812032734833212, 1.69405882352552328190478774719, 1.82253461326485157326253031006, 2.08178636012757952688851218408, 2.63412286386940625106858155014, 2.99267196603154114743720235527, 3.03652081732347157195545113311, 3.48617716682372907270574098783, 3.66938354277451455100568008582, 3.83757511907556622262080628487, 4.15662587948371313749634278917, 4.47625231756729024719877778905, 4.49454858810118512447501815079, 5.08897951785586896716068707039, 5.16264770486478922947440877943, 5.27288193137734556227974655274, 5.73214210635757272560599836299, 5.77927732917681737026692716288, 5.91868047272880425165128077794, 6.20834011992144226178335123324, 6.31114603329833797337362154568, 6.67737889969777734493315144302, 6.97571308322249683287413505120
