| L(s) = 1 | + 3·3-s + 3·7-s + 6·9-s + 6·19-s + 9·21-s + 10·27-s − 12·29-s + 3·31-s − 6·37-s + 21·43-s − 12·47-s + 6·49-s − 6·53-s + 18·57-s + 6·59-s + 3·61-s + 18·63-s + 27·67-s + 12·71-s + 9·73-s − 9·79-s + 15·81-s − 18·83-s − 36·87-s − 24·89-s + 9·93-s − 21·97-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.13·7-s + 2·9-s + 1.37·19-s + 1.96·21-s + 1.92·27-s − 2.22·29-s + 0.538·31-s − 0.986·37-s + 3.20·43-s − 1.75·47-s + 6/7·49-s − 0.824·53-s + 2.38·57-s + 0.781·59-s + 0.384·61-s + 2.26·63-s + 3.29·67-s + 1.42·71-s + 1.05·73-s − 1.01·79-s + 5/3·81-s − 1.97·83-s − 3.85·87-s − 2.54·89-s + 0.933·93-s − 2.13·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.62566053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.62566053\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $D_{6}$ | \( 1 - 2 T^{3} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 3 T^{2} - 6 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 8 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 30 T^{2} + 16 T^{3} + 30 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 21 T^{2} - 20 T^{3} + 21 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 45 T^{2} - 8 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 12 T + 120 T^{2} + 702 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 81 T^{2} - 170 T^{3} + 81 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 102 T^{2} + 426 T^{3} + 102 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 66 T^{2} - 156 T^{3} + 66 p T^{4} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 21 T + 255 T^{2} - 2018 T^{3} + 255 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 165 T^{2} + 1104 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 428 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 676 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 138 T^{2} - 199 T^{3} + 138 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 27 T + 423 T^{2} - 4142 T^{3} + 423 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 237 T^{2} - 1664 T^{3} + 237 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 186 T^{2} - 1145 T^{3} + 186 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 1438 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 237 T^{2} + 1996 T^{3} + 237 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 24 T + 399 T^{2} + 4320 T^{3} + 399 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 21 T + 423 T^{2} + 4310 T^{3} + 423 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.12404639569665187877935060491, −6.72852343416445593432092816184, −6.66083898543128818275623930313, −6.33474474586811010497403058514, −5.77814599739960985306150567832, −5.62722860431681652715986995653, −5.56150949518056302223046549025, −5.17577021097293324623786237849, −5.17082425503951890388431932340, −4.72850975996421025499019549823, −4.27881130825608904109717942493, −4.23153084687601711150844864763, −4.11237685736333142575421111520, −3.69398370866153121224039493738, −3.47363584932659434163518151631, −3.29596905097072717731036906283, −2.84163844020425474232110561602, −2.70764088629755819356285101718, −2.47059240046677926020914839401, −1.99830498035532661553437333961, −1.73821565662767161778269891108, −1.72420577059930845981412376274, −1.22502617332601423812403138027, −0.77662766392430302970864618650, −0.50580468635573978042579600483,
0.50580468635573978042579600483, 0.77662766392430302970864618650, 1.22502617332601423812403138027, 1.72420577059930845981412376274, 1.73821565662767161778269891108, 1.99830498035532661553437333961, 2.47059240046677926020914839401, 2.70764088629755819356285101718, 2.84163844020425474232110561602, 3.29596905097072717731036906283, 3.47363584932659434163518151631, 3.69398370866153121224039493738, 4.11237685736333142575421111520, 4.23153084687601711150844864763, 4.27881130825608904109717942493, 4.72850975996421025499019549823, 5.17082425503951890388431932340, 5.17577021097293324623786237849, 5.56150949518056302223046549025, 5.62722860431681652715986995653, 5.77814599739960985306150567832, 6.33474474586811010497403058514, 6.66083898543128818275623930313, 6.72852343416445593432092816184, 7.12404639569665187877935060491
