| L(s) = 1 | − 2·7-s + 12·19-s + 6·25-s − 8·29-s + 12·37-s − 8·47-s + 6·49-s + 16·53-s − 16·59-s + 8·83-s + 32·103-s − 16·113-s + 34·121-s + 127-s + 131-s − 24·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s − 12·175-s + 179-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2.75·19-s + 6/5·25-s − 1.48·29-s + 1.97·37-s − 1.16·47-s + 6/7·49-s + 2.19·53-s − 2.08·59-s + 0.878·83-s + 3.15·103-s − 1.50·113-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s − 0.907·175-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.150534770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.150534770\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 22 T^{2} + 682 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 82 T^{2} + 2722 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 102 T^{2} + 5130 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 24 T^{2} + 3774 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 144 T^{2} + 10830 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 114 T^{2} + 8418 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 236 T^{2} + 24310 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 288 T^{2} + 33150 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 294 T^{2} + 36618 T^{4} - 294 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.83532392847784939186540283921, −5.75029890065780653970207647440, −5.73707824961660140851325684396, −5.30132914525502660566795309341, −5.19050717702498478032606852035, −4.96986241335451230661389005502, −4.77345834966817422309629118918, −4.76580770897258063513373201968, −4.33817469635138339313694855890, −4.10787661978454326185073248864, −3.88774947572186862248852957608, −3.69571964905591572920396265353, −3.40590503258633021842193901971, −3.29718623508655407620193637815, −3.26254585231518776266882967569, −2.66805334508012041709823063983, −2.61442834826053004758013146829, −2.59114881439290876133384350252, −2.20348813789904254428360735070, −1.69955911297611274664839973417, −1.62636631647181878795950817625, −1.10820382952190459478350692553, −1.01854328092466207080362830095, −0.74642937992766200345474028665, −0.21555920372981591326672627010,
0.21555920372981591326672627010, 0.74642937992766200345474028665, 1.01854328092466207080362830095, 1.10820382952190459478350692553, 1.62636631647181878795950817625, 1.69955911297611274664839973417, 2.20348813789904254428360735070, 2.59114881439290876133384350252, 2.61442834826053004758013146829, 2.66805334508012041709823063983, 3.26254585231518776266882967569, 3.29718623508655407620193637815, 3.40590503258633021842193901971, 3.69571964905591572920396265353, 3.88774947572186862248852957608, 4.10787661978454326185073248864, 4.33817469635138339313694855890, 4.76580770897258063513373201968, 4.77345834966817422309629118918, 4.96986241335451230661389005502, 5.19050717702498478032606852035, 5.30132914525502660566795309341, 5.73707824961660140851325684396, 5.75029890065780653970207647440, 5.83532392847784939186540283921
