| L(s) = 1 | + 3·2-s + 6·4-s + 3·5-s − 3·7-s + 10·8-s + 9·10-s − 3·11-s − 6·13-s − 9·14-s + 15·16-s − 4·17-s − 2·19-s + 18·20-s − 9·22-s − 8·23-s + 6·25-s − 18·26-s − 18·28-s − 10·29-s + 21·32-s − 12·34-s − 9·35-s − 8·37-s − 6·38-s + 30·40-s − 6·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 1.34·5-s − 1.13·7-s + 3.53·8-s + 2.84·10-s − 0.904·11-s − 1.66·13-s − 2.40·14-s + 15/4·16-s − 0.970·17-s − 0.458·19-s + 4.02·20-s − 1.91·22-s − 1.66·23-s + 6/5·25-s − 3.53·26-s − 3.40·28-s − 1.85·29-s + 3.71·32-s − 2.05·34-s − 1.52·35-s − 1.31·37-s − 0.973·38-s + 4.74·40-s − 0.937·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 13 | $A_4\times C_2$ | \( 1 + 6 T + 23 T^{2} + 4 p T^{3} + 23 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 4 T - 9 T^{2} - 96 T^{3} - 9 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 2 T + 21 T^{2} + 68 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 8 T + 81 T^{2} + 360 T^{3} + 81 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 10 T + 83 T^{2} + 476 T^{3} + 83 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 37 | $A_4\times C_2$ | \( 1 + 8 T + 67 T^{2} + 248 T^{3} + 67 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 6 T + 23 T^{2} - 172 T^{3} + 23 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 10 T + 125 T^{2} + 756 T^{3} + 125 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 2 T + 105 T^{2} + 180 T^{3} + 105 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 2 T + 11 T^{2} + 20 T^{3} + 11 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 8 T + 161 T^{2} - 880 T^{3} + 161 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 16 T + 231 T^{2} + 1888 T^{3} + 231 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 18 T + 225 T^{2} + 2068 T^{3} + 225 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $A_4\times C_2$ | \( 1 + 22 T + 343 T^{2} + 3316 T^{3} + 343 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 125 T^{2} - 448 T^{3} + 125 p T^{4} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 137 T^{2} - 448 T^{3} + 137 p T^{4} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 18 T + 347 T^{2} + 3196 T^{3} + 347 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 10 T + 287 T^{2} + 1836 T^{3} + 287 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.29985202080248611945799813591, −6.84638053117058738485902203539, −6.83650015239775037156601699236, −6.79429449537625303957198959045, −6.11819734272799287818514877468, −6.11173512040594151699632530782, −6.05639059859206958428626481538, −5.62119223331644699582623609091, −5.49374779815302578928346187513, −5.35090999385251935083630206286, −4.95332433040307278298442848881, −4.71157664064781647726304194961, −4.69100651200230477145683472526, −4.18439354041992720973330580377, −3.95898432363682501830214545106, −3.92990059044975305845978612811, −3.30708670697233897510091212473, −3.19702678613482012337103444041, −2.93588352340358664723322917270, −2.59366420017884064778835030065, −2.40680128160189106919662280077, −2.27763099619327749482762829578, −1.63315868795586370433664844648, −1.56965619957195072456527263175, −1.52773438962417333707983868816, 0, 0, 0,
1.52773438962417333707983868816, 1.56965619957195072456527263175, 1.63315868795586370433664844648, 2.27763099619327749482762829578, 2.40680128160189106919662280077, 2.59366420017884064778835030065, 2.93588352340358664723322917270, 3.19702678613482012337103444041, 3.30708670697233897510091212473, 3.92990059044975305845978612811, 3.95898432363682501830214545106, 4.18439354041992720973330580377, 4.69100651200230477145683472526, 4.71157664064781647726304194961, 4.95332433040307278298442848881, 5.35090999385251935083630206286, 5.49374779815302578928346187513, 5.62119223331644699582623609091, 6.05639059859206958428626481538, 6.11173512040594151699632530782, 6.11819734272799287818514877468, 6.79429449537625303957198959045, 6.83650015239775037156601699236, 6.84638053117058738485902203539, 7.29985202080248611945799813591
