| L(s) = 1 | − 3·3-s − 3·5-s − 3·7-s + 6·9-s + 11-s − 11·13-s + 9·15-s + 5·19-s + 9·21-s + 3·23-s − 5·25-s − 10·27-s − 4·29-s − 2·31-s − 3·33-s + 9·35-s − 8·37-s + 33·39-s − 11·41-s + 11·43-s − 18·45-s − 47-s + 6·49-s − 4·53-s − 3·55-s − 15·57-s − 10·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 1.13·7-s + 2·9-s + 0.301·11-s − 3.05·13-s + 2.32·15-s + 1.14·19-s + 1.96·21-s + 0.625·23-s − 25-s − 1.92·27-s − 0.742·29-s − 0.359·31-s − 0.522·33-s + 1.52·35-s − 1.31·37-s + 5.28·39-s − 1.71·41-s + 1.67·43-s − 2.68·45-s − 0.145·47-s + 6/7·49-s − 0.549·53-s − 0.404·55-s − 1.98·57-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4282430282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4282430282\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 3 T + 14 T^{2} + 26 T^{3} + 14 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 12 T^{2} + 15 T^{3} + 12 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11 T + 74 T^{2} + 24 p T^{3} + 74 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T^{2} - 52 T^{3} - 6 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 5 T + 40 T^{2} - 137 T^{3} + 40 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 80 T^{2} + 206 T^{3} + 80 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 44 T^{2} + 222 T^{3} + 44 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 100 T^{2} + 578 T^{3} + 100 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 11 T + 138 T^{2} + 843 T^{3} + 138 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 11 T + 142 T^{2} - 950 T^{3} + 142 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + T + 91 T^{2} - 54 T^{3} + 91 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 4 T + 137 T^{2} + 335 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 129 T^{2} + 699 T^{3} + 129 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 22 T + 313 T^{2} + 2857 T^{3} + 313 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 11 T + 164 T^{2} - 978 T^{3} + 164 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 158 T^{2} - 1166 T^{3} + 158 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 220 T^{2} + 1336 T^{3} + 220 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 188 T^{2} + 414 T^{3} + 188 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 214 T^{2} - 1758 T^{3} + 214 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 9 T + 218 T^{2} + 1604 T^{3} + 218 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T - 10 T^{2} + 282 T^{3} - 10 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.11788742556150767740521496285, −6.64954586759819295816946448722, −6.49719310644839288268329570448, −6.41907459066623246056143349431, −5.79733422192656310159452453613, −5.72768604910379097792206463219, −5.72161546273651507859515959431, −5.16975667302840806631014138896, −5.02549683571247588508565734830, −5.01604833175230412863727076537, −4.49879825643089357177514200720, −4.34907105107315172499868487063, −4.29770708988595098514945770371, −3.63940091199571034108160275026, −3.59636705090905288100565436367, −3.48998545373617690878845897184, −2.91265446313703771091051568341, −2.76487463325331296131515215752, −2.56937446840366153119703219927, −1.77079974024357524458417233973, −1.76870898015729211768150745502, −1.59002199976344714893013399483, −0.63087884767814348039292938058, −0.43744800029521018025853051920, −0.31123115992331935343446062897,
0.31123115992331935343446062897, 0.43744800029521018025853051920, 0.63087884767814348039292938058, 1.59002199976344714893013399483, 1.76870898015729211768150745502, 1.77079974024357524458417233973, 2.56937446840366153119703219927, 2.76487463325331296131515215752, 2.91265446313703771091051568341, 3.48998545373617690878845897184, 3.59636705090905288100565436367, 3.63940091199571034108160275026, 4.29770708988595098514945770371, 4.34907105107315172499868487063, 4.49879825643089357177514200720, 5.01604833175230412863727076537, 5.02549683571247588508565734830, 5.16975667302840806631014138896, 5.72161546273651507859515959431, 5.72768604910379097792206463219, 5.79733422192656310159452453613, 6.41907459066623246056143349431, 6.49719310644839288268329570448, 6.64954586759819295816946448722, 7.11788742556150767740521496285
