| L(s) = 1 | − 3·5-s − 2·11-s − 5·13-s + 2·17-s + 19-s + 6·23-s + 6·25-s + 12·29-s − 3·31-s − 9·37-s − 10·41-s − 43-s − 10·47-s + 4·53-s + 6·55-s + 16·59-s + 2·61-s + 15·65-s − 5·67-s + 20·71-s − 15·73-s − 13·79-s + 2·83-s − 6·85-s − 2·89-s − 3·95-s − 12·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.603·11-s − 1.38·13-s + 0.485·17-s + 0.229·19-s + 1.25·23-s + 6/5·25-s + 2.22·29-s − 0.538·31-s − 1.47·37-s − 1.56·41-s − 0.152·43-s − 1.45·47-s + 0.549·53-s + 0.809·55-s + 2.08·59-s + 0.256·61-s + 1.86·65-s − 0.610·67-s + 2.37·71-s − 1.75·73-s − 1.46·79-s + 0.219·83-s − 0.650·85-s − 0.211·89-s − 0.307·95-s − 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.564684790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.564684790\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | | \( 1 \) |
| good | 11 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} + 62 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 28 T^{2} + 133 T^{3} + 28 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 86 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - T + 40 T^{2} - 17 T^{3} + 40 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 33 T^{2} - 114 T^{3} + 33 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 12 T + 69 T^{2} - 318 T^{3} + 69 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 + 9 T + 90 T^{2} + 475 T^{3} + 90 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 45 T^{2} + 46 T^{3} + 45 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + T + 104 T^{2} + p T^{3} + 104 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 105 T^{2} + 868 T^{3} + 105 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 63 T^{2} - 568 T^{3} + 63 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 16 T + 243 T^{2} - 1906 T^{3} + 243 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 167 T^{2} - 248 T^{3} + 167 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 5 T + 190 T^{2} + 673 T^{3} + 190 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 20 T + 327 T^{2} - 3038 T^{3} + 327 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 15 T + 246 T^{2} + 2149 T^{3} + 246 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 13 T + 128 T^{2} + 601 T^{3} + 128 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 93 T^{2} - 854 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 230 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $D_{6}$ | \( 1 + 12 T + 267 T^{2} + 2212 T^{3} + 267 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.94348249264826125528501009759, −6.77088584990498875978858640366, −6.44979565394900555905164345144, −6.33034246355333686056304392217, −5.92938374244420199883254682641, −5.45232945774816732373569672081, −5.27992991243703387087997354217, −5.23579605656082713731081416080, −5.08838194564283236594058058263, −4.74262400489857696547608803200, −4.49464208910651960342705194981, −4.28208416392592566966779702753, −4.04457300681858860027579097998, −3.56563087332397534574548657431, −3.54190713587186295648452430324, −3.18508122873793967674685589087, −2.79337380892110283048586667652, −2.79190048783959710160481967504, −2.65726954848364897683588097619, −1.88293442085013186933000536540, −1.72411042689630298921012040838, −1.60054286370532414661996942528, −0.74147864684393060115654034015, −0.64907726058610763378633131848, −0.38217644204947088472056642918,
0.38217644204947088472056642918, 0.64907726058610763378633131848, 0.74147864684393060115654034015, 1.60054286370532414661996942528, 1.72411042689630298921012040838, 1.88293442085013186933000536540, 2.65726954848364897683588097619, 2.79190048783959710160481967504, 2.79337380892110283048586667652, 3.18508122873793967674685589087, 3.54190713587186295648452430324, 3.56563087332397534574548657431, 4.04457300681858860027579097998, 4.28208416392592566966779702753, 4.49464208910651960342705194981, 4.74262400489857696547608803200, 5.08838194564283236594058058263, 5.23579605656082713731081416080, 5.27992991243703387087997354217, 5.45232945774816732373569672081, 5.92938374244420199883254682641, 6.33034246355333686056304392217, 6.44979565394900555905164345144, 6.77088584990498875978858640366, 6.94348249264826125528501009759
