| L(s) = 1 | − 3·5-s − 5·7-s + 2·11-s − 2·17-s − 4·19-s + 7·23-s + 6·25-s + 7·29-s − 16·31-s + 15·35-s + 2·37-s + 41-s + 2·43-s + 13·47-s + 7·49-s + 10·53-s − 6·55-s + 6·59-s − 11·61-s + 67-s + 14·71-s + 16·73-s − 10·77-s − 6·79-s + 21·83-s + 6·85-s + 33·89-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.88·7-s + 0.603·11-s − 0.485·17-s − 0.917·19-s + 1.45·23-s + 6/5·25-s + 1.29·29-s − 2.87·31-s + 2.53·35-s + 0.328·37-s + 0.156·41-s + 0.304·43-s + 1.89·47-s + 49-s + 1.37·53-s − 0.809·55-s + 0.781·59-s − 1.40·61-s + 0.122·67-s + 1.66·71-s + 1.87·73-s − 1.13·77-s − 0.675·79-s + 2.30·83-s + 0.650·85-s + 3.49·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.148755577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148755577\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T + 18 T^{2} + 61 T^{3} + 18 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 25 T^{2} - 32 T^{3} + 25 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 15 T^{2} + 36 T^{3} + 15 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} - 40 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 148 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 18 T^{2} - 19 T^{3} + 18 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 82 T^{2} - 379 T^{3} + 82 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 16 T + 169 T^{2} + 1100 T^{3} + 169 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 103 T^{2} - 136 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 114 T^{2} - 85 T^{3} + 114 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 93 T^{2} - 64 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 13 T + 56 T^{2} - 99 T^{3} + 56 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 171 T^{2} - 1036 T^{3} + 171 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 117 T^{2} - 780 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 11 T + 142 T^{2} + 823 T^{3} + 142 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + 68 T^{2} - 247 T^{3} + 68 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 14 T + 193 T^{2} - 1952 T^{3} + 193 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 1952 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 153 T^{2} + 1052 T^{3} + 153 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 21 T + 378 T^{2} - 3729 T^{3} + 378 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 - 30 T + 447 T^{2} - 4724 T^{3} + 447 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.84167275637524187743179578794, −7.28039958880379925094293282380, −7.11700818745163406138948595023, −6.99229446313867389525283407501, −6.67350916231201098957082043365, −6.46999931018072146904640354411, −6.28367039274199625422241268066, −5.81984726395999970787916651233, −5.81188232542470481630340351539, −5.43199846600304346592408594146, −4.74840660525938939686854171622, −4.74772498809632743021500900292, −4.74578755427791903554201022641, −4.15486022797366647899467277376, −3.72644720504476567311108355054, −3.69248209589263663245278266303, −3.40629437780267487810522350806, −3.22451169527086838302076796577, −2.94295785311676207777245124738, −2.22183597299868185875472081513, −2.14446588557025644013056409854, −1.96167084257878982962335044628, −0.807783986212037070958236308939, −0.75466189941054143120195042745, −0.49751083740021633078767928261,
0.49751083740021633078767928261, 0.75466189941054143120195042745, 0.807783986212037070958236308939, 1.96167084257878982962335044628, 2.14446588557025644013056409854, 2.22183597299868185875472081513, 2.94295785311676207777245124738, 3.22451169527086838302076796577, 3.40629437780267487810522350806, 3.69248209589263663245278266303, 3.72644720504476567311108355054, 4.15486022797366647899467277376, 4.74578755427791903554201022641, 4.74772498809632743021500900292, 4.74840660525938939686854171622, 5.43199846600304346592408594146, 5.81188232542470481630340351539, 5.81984726395999970787916651233, 6.28367039274199625422241268066, 6.46999931018072146904640354411, 6.67350916231201098957082043365, 6.99229446313867389525283407501, 7.11700818745163406138948595023, 7.28039958880379925094293282380, 7.84167275637524187743179578794
