| L(s) = 1 | + 3·3-s + 3·7-s + 6·9-s + 2·11-s + 12·13-s + 3·19-s + 9·21-s + 2·23-s − 5·25-s + 10·27-s + 2·29-s + 8·31-s + 6·33-s + 6·37-s + 36·39-s + 2·41-s + 4·43-s + 4·47-s + 6·49-s − 14·53-s + 9·57-s − 4·59-s + 2·61-s + 18·63-s − 2·67-s + 6·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.13·7-s + 2·9-s + 0.603·11-s + 3.32·13-s + 0.688·19-s + 1.96·21-s + 0.417·23-s − 25-s + 1.92·27-s + 0.371·29-s + 1.43·31-s + 1.04·33-s + 0.986·37-s + 5.76·39-s + 0.312·41-s + 0.609·43-s + 0.583·47-s + 6/7·49-s − 1.92·53-s + 1.19·57-s − 0.520·59-s + 0.256·61-s + 2.26·63-s − 0.244·67-s + 0.722·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(25.31121787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(25.31121787\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 + p T^{2} - 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 9 T^{2} - 60 T^{3} + 9 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 41 T^{2} - 8 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 45 T^{2} - 108 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 69 T^{2} - 72 T^{3} + 69 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 89 T^{2} - 480 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 47 T^{2} - 364 T^{3} + 47 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 109 T^{2} - 280 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 127 T^{2} - 384 T^{3} + 127 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 173 T^{2} + 1480 T^{3} + 173 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 216 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 51 T^{2} - 236 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 177 T^{2} + 284 T^{3} + 177 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 167 T^{2} + 1120 T^{3} + 167 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 14 T + 207 T^{2} - 2036 T^{3} + 207 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 205 T^{2} + 284 T^{3} + 205 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 187 T^{2} + 432 T^{3} + 187 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 223 T^{2} - 1308 T^{3} + 223 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 519 T^{2} - 5944 T^{3} + 519 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.38463198361793927193907405305, −6.71106048215343973932414234318, −6.63080931970794129547440742368, −6.39126697049870655847126734062, −6.22630080984453202101342955366, −6.00953123859896696097674532888, −5.78655166680138612952879652735, −5.27210361321342749738153137376, −5.24966591559383968788927016045, −4.78749299024637643212356796634, −4.48545106935391086806280911879, −4.36110518447050561736124860025, −4.05168196980367434060395439931, −3.85297955607231846066452191465, −3.57399395075995681536588629263, −3.41374180330353675539684509938, −3.06538824690651378290260032082, −2.72354441751629329838829069511, −2.66853276329211457540927668010, −2.00202384973988332764802840002, −1.77815030923714473994012239929, −1.65936719572374694223231835700, −1.16460905869861157777417648170, −0.904788374363327298438206443996, −0.75402570218372727066863855989,
0.75402570218372727066863855989, 0.904788374363327298438206443996, 1.16460905869861157777417648170, 1.65936719572374694223231835700, 1.77815030923714473994012239929, 2.00202384973988332764802840002, 2.66853276329211457540927668010, 2.72354441751629329838829069511, 3.06538824690651378290260032082, 3.41374180330353675539684509938, 3.57399395075995681536588629263, 3.85297955607231846066452191465, 4.05168196980367434060395439931, 4.36110518447050561736124860025, 4.48545106935391086806280911879, 4.78749299024637643212356796634, 5.24966591559383968788927016045, 5.27210361321342749738153137376, 5.78655166680138612952879652735, 6.00953123859896696097674532888, 6.22630080984453202101342955366, 6.39126697049870655847126734062, 6.63080931970794129547440742368, 6.71106048215343973932414234318, 7.38463198361793927193907405305
