| L(s) = 1 | − 3·3-s − 6·7-s + 6·9-s − 7·11-s + 3·17-s − 13·19-s + 18·21-s + 14·23-s − 8·25-s − 10·27-s + 10·29-s − 6·31-s + 21·33-s + 6·37-s + 13·41-s + 5·43-s − 7·47-s + 10·49-s − 9·51-s − 3·53-s + 39·57-s + 6·59-s − 61-s − 36·63-s − 33·67-s − 42·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.26·7-s + 2·9-s − 2.11·11-s + 0.727·17-s − 2.98·19-s + 3.92·21-s + 2.91·23-s − 8/5·25-s − 1.92·27-s + 1.85·29-s − 1.07·31-s + 3.65·33-s + 0.986·37-s + 2.03·41-s + 0.762·43-s − 1.02·47-s + 10/7·49-s − 1.26·51-s − 0.412·53-s + 5.16·57-s + 0.781·59-s − 0.128·61-s − 4.53·63-s − 4.03·67-s − 5.05·69-s + 0.712·71-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 8 T^{2} + 7 T^{3} + 8 p T^{4} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 6 T + 26 T^{2} + 85 T^{3} + 26 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 7 T + 47 T^{2} + 161 T^{3} + 47 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 3 T + 47 T^{2} - 103 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 13 T + 97 T^{2} + 481 T^{3} + 97 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 14 T + 118 T^{2} - 651 T^{3} + 118 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 10 T + 90 T^{2} - 567 T^{3} + 90 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 6 T + 56 T^{2} + 191 T^{3} + 56 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 6 T + 2 p T^{2} - 445 T^{3} + 2 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 13 T + 149 T^{2} - 983 T^{3} + 149 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 5 T + 100 T^{2} - 389 T^{3} + 100 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 7 T + 85 T^{2} + 357 T^{3} + 85 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} - 325 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T - 7 T^{2} + 404 T^{3} - 7 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + T + 111 T^{2} + 163 T^{3} + 111 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 33 T + 557 T^{2} + 5669 T^{3} + 557 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 6 T + 162 T^{2} - 13 p T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 14 T + 212 T^{2} - 1953 T^{3} + 212 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 11 T + 261 T^{2} - 1751 T^{3} + 261 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 8 T + 100 T^{2} + 1299 T^{3} + 100 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 7 T + 141 T^{2} + 273 T^{3} + 141 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 19 T + 346 T^{2} - 3715 T^{3} + 346 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.37022011004766787251642949141, −6.71981194587918788290424786158, −6.66975396196763504237313383110, −6.59077038303723228623100234776, −6.31025218374087677389105812764, −6.12844539490820135952952803904, −6.02188160081052329649926915263, −5.51250952524918086137506144380, −5.42934869667619169002760799221, −5.39023774707767006831233026496, −4.80648460225367436928455319423, −4.79308798146421432793064241491, −4.57118795956099656753855827238, −4.00674054661538619436325918595, −3.98038570135656662648328815725, −3.92167384996497426746763322221, −3.11019345331780987540545319888, −3.05080134041651731827755956288, −2.98094012196096485790552932787, −2.55131771279509385537543670759, −2.23632890725472614767458600096, −2.12191129076108914383486919264, −1.28705818829591757926300790777, −1.20508496717562071007319007153, −0.72925221299194677698636624754, 0, 0, 0,
0.72925221299194677698636624754, 1.20508496717562071007319007153, 1.28705818829591757926300790777, 2.12191129076108914383486919264, 2.23632890725472614767458600096, 2.55131771279509385537543670759, 2.98094012196096485790552932787, 3.05080134041651731827755956288, 3.11019345331780987540545319888, 3.92167384996497426746763322221, 3.98038570135656662648328815725, 4.00674054661538619436325918595, 4.57118795956099656753855827238, 4.79308798146421432793064241491, 4.80648460225367436928455319423, 5.39023774707767006831233026496, 5.42934869667619169002760799221, 5.51250952524918086137506144380, 6.02188160081052329649926915263, 6.12844539490820135952952803904, 6.31025218374087677389105812764, 6.59077038303723228623100234776, 6.66975396196763504237313383110, 6.71981194587918788290424786158, 7.37022011004766787251642949141
