| L(s) = 1 | − 3-s − 3·5-s − 7-s − 6·9-s + 5·11-s + 3·15-s − 17-s + 21-s − 4·23-s + 6·25-s + 8·27-s − 2·29-s − 7·31-s − 5·33-s + 3·35-s + 9·37-s + 8·41-s − 43-s + 18·45-s + 10·47-s − 18·49-s + 51-s − 8·53-s − 15·55-s − 17·59-s − 29·61-s + 6·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s − 2·9-s + 1.50·11-s + 0.774·15-s − 0.242·17-s + 0.218·21-s − 0.834·23-s + 6/5·25-s + 1.53·27-s − 0.371·29-s − 1.25·31-s − 0.870·33-s + 0.507·35-s + 1.47·37-s + 1.24·41-s − 0.152·43-s + 2.68·45-s + 1.45·47-s − 2.57·49-s + 0.140·51-s − 1.09·53-s − 2.02·55-s − 2.21·59-s − 3.71·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $A_4\times C_2$ | \( 1 + T + 7 T^{2} + 5 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + T + 19 T^{2} + 13 T^{3} + 19 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 39 T^{2} - 111 T^{3} + 39 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 50 T^{2} + 7 T^{3} + 50 p T^{4} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 4 T + 72 T^{2} + 183 T^{3} + 72 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 2 T + 72 T^{2} + 129 T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 7 T + 3 p T^{2} + 427 T^{3} + 3 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 41 | $A_4\times C_2$ | \( 1 - 8 T + 114 T^{2} - 543 T^{3} + 114 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + T + 85 T^{2} + 169 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 10 T + 88 T^{2} - 381 T^{3} + 88 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 8 T + 80 T^{2} + 679 T^{3} + 80 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 17 T + 208 T^{2} + 1965 T^{3} + 208 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 29 T + 461 T^{2} + 4419 T^{3} + 461 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 17 T + 260 T^{2} - 2265 T^{3} + 260 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 3 T + 195 T^{2} + 413 T^{3} + 195 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 16 T + 239 T^{2} - 2328 T^{3} + 239 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 13 T + 179 T^{2} + 1257 T^{3} + 179 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 17 T + 259 T^{2} + 2235 T^{3} + 259 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 9 T + 98 T^{2} - 649 T^{3} + 98 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 6 T + 170 T^{2} - 997 T^{3} + 170 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.129524860351160967602615589161, −7.66221795543747090776655959796, −7.65375518663501261254172766489, −7.38935283700206309246520315499, −6.90657784097339399148054081145, −6.63366444798710717289948170631, −6.43917760151076638664330406770, −6.24301732697016904054401237717, −5.99123918238279848715943857648, −5.96264377807581054154092373009, −5.36325975673600152621247711199, −5.18869562471565287705091095634, −5.10018862679270079641716344367, −4.39573132157882191165078207005, −4.33739185489403060948312598335, −4.26239835951144055000649331144, −3.60199541478232169706213200881, −3.51475223178444634711757645483, −3.45826482512399454222973601199, −2.66495361379809537651033610722, −2.63832532102359650479559958733, −2.58810917836808630316602048381, −1.65551902708036849538216816262, −1.33938636205191308093943219937, −1.12647157484703852866612872388, 0, 0, 0,
1.12647157484703852866612872388, 1.33938636205191308093943219937, 1.65551902708036849538216816262, 2.58810917836808630316602048381, 2.63832532102359650479559958733, 2.66495361379809537651033610722, 3.45826482512399454222973601199, 3.51475223178444634711757645483, 3.60199541478232169706213200881, 4.26239835951144055000649331144, 4.33739185489403060948312598335, 4.39573132157882191165078207005, 5.10018862679270079641716344367, 5.18869562471565287705091095634, 5.36325975673600152621247711199, 5.96264377807581054154092373009, 5.99123918238279848715943857648, 6.24301732697016904054401237717, 6.43917760151076638664330406770, 6.63366444798710717289948170631, 6.90657784097339399148054081145, 7.38935283700206309246520315499, 7.65375518663501261254172766489, 7.66221795543747090776655959796, 8.129524860351160967602615589161
