| L(s) = 1 | − 3·2-s + 6·4-s − 5-s − 10·8-s + 3·10-s + 11-s − 8·13-s + 15·16-s + 4·17-s + 3·19-s − 6·20-s − 3·22-s + 7·23-s − 8·25-s + 24·26-s + 5·29-s − 20·31-s − 21·32-s − 12·34-s − 3·37-s − 9·38-s + 10·40-s + 6·43-s + 6·44-s − 21·46-s + 9·47-s + 24·50-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s − 0.447·5-s − 3.53·8-s + 0.948·10-s + 0.301·11-s − 2.21·13-s + 15/4·16-s + 0.970·17-s + 0.688·19-s − 1.34·20-s − 0.639·22-s + 1.45·23-s − 8/5·25-s + 4.70·26-s + 0.928·29-s − 3.59·31-s − 3.71·32-s − 2.05·34-s − 0.493·37-s − 1.45·38-s + 1.58·40-s + 0.914·43-s + 0.904·44-s − 3.09·46-s + 1.31·47-s + 3.39·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} + 13 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 27 T^{2} - 25 T^{3} + 27 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 40 T^{2} + 139 T^{3} + 40 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 39 T^{2} - 112 T^{3} + 39 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 65 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 81 T^{2} - 325 T^{3} + 81 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 21 T^{2} + 73 T^{3} + 21 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 20 T + 214 T^{2} + 1441 T^{3} + 214 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 90 T^{2} + 9 T^{3} + 90 p T^{4} + p^{3} T^{6} \) |
| 43 | $D_{6}$ | \( 1 - 6 T + 60 T^{2} - 389 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 9 T + 87 T^{2} - 657 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 15 T + 225 T^{2} + 1671 T^{3} + 225 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 216 T^{2} - 1589 T^{3} + 216 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 178 T^{2} + 883 T^{3} + 178 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 89 T^{2} - 77 T^{3} + 89 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 7 T + 15 T^{2} + 599 T^{3} + 15 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 19 T + 227 T^{2} + 2143 T^{3} + 227 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 5 T + 163 T^{2} + 469 T^{3} + 163 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 186 T^{2} + 185 T^{3} + 186 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 9 T + 225 T^{2} - 1611 T^{3} + 225 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 28 T + 503 T^{2} + 5680 T^{3} + 503 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.46880715031899841783983785202, −7.14033738396788928223952355078, −7.06640726301080162512418876965, −6.95490711506134705828611813543, −6.44528600748073401679346288212, −6.09294115403403593941476108472, −6.02880602305190697724527510868, −5.71465271940635416806878967742, −5.44991922325104677414759527496, −5.28761221816122638832749231025, −5.00425916402334706879225779128, −4.66156076877008729972728394940, −4.58270653187333127233097840671, −3.93174502317013269266404818489, −3.76965846253541146368740550458, −3.67331377740717930902660103139, −3.17019298003706518576796864164, −2.94131041764567863745549604474, −2.86722500313054288476883768679, −2.15617317862415885789292865547, −2.14240871843663363036247659392, −2.10298090273673655530903502486, −1.27976717202484886101414412246, −1.22480982461485297221021259823, −0.990424096893601264252266618152, 0, 0, 0,
0.990424096893601264252266618152, 1.22480982461485297221021259823, 1.27976717202484886101414412246, 2.10298090273673655530903502486, 2.14240871843663363036247659392, 2.15617317862415885789292865547, 2.86722500313054288476883768679, 2.94131041764567863745549604474, 3.17019298003706518576796864164, 3.67331377740717930902660103139, 3.76965846253541146368740550458, 3.93174502317013269266404818489, 4.58270653187333127233097840671, 4.66156076877008729972728394940, 5.00425916402334706879225779128, 5.28761221816122638832749231025, 5.44991922325104677414759527496, 5.71465271940635416806878967742, 6.02880602305190697724527510868, 6.09294115403403593941476108472, 6.44528600748073401679346288212, 6.95490711506134705828611813543, 7.06640726301080162512418876965, 7.14033738396788928223952355078, 7.46880715031899841783983785202
