| L(s) = 1 | + 4·2-s + 2·4-s − 3·5-s − 20·8-s − 12·10-s − 45·16-s − 8·17-s − 6·20-s + 25-s + 16·32-s − 32·34-s − 12·37-s + 60·40-s − 30·43-s − 22·47-s + 4·49-s + 4·50-s − 12·59-s + 204·64-s − 16·68-s − 24·71-s + 12·73-s − 48·74-s + 135·80-s + 24·85-s − 120·86-s − 88·94-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4-s − 1.34·5-s − 7.07·8-s − 3.79·10-s − 11.2·16-s − 1.94·17-s − 1.34·20-s + 1/5·25-s + 2.82·32-s − 5.48·34-s − 1.97·37-s + 9.48·40-s − 4.57·43-s − 3.20·47-s + 4/7·49-s + 0.565·50-s − 1.56·59-s + 51/2·64-s − 1.94·68-s − 2.84·71-s + 1.40·73-s − 5.57·74-s + 15.0·80-s + 2.60·85-s − 12.9·86-s − 9.07·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3213810469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3213810469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 61 T^{2} + 1896 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 3534 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 185 T^{2} + 14136 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3870 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6606 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 6360 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 248 T^{2} + 30606 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.65246216108314240589369818666, −6.55530885698656436411880909652, −6.51569285147248066793496773061, −6.11339804710674553480174302421, −5.73915444991662464568455798372, −5.66939374135429044815198702286, −5.32241656275636039305172397398, −5.11962945969051842312858822497, −5.11838058514613447582508660963, −4.74999179007308745763572296139, −4.58259226043927656285763485837, −4.41564703159337459781153452433, −4.23765173728633795794916638737, −4.11123116851441865629012377638, −3.75756267574416350282101829013, −3.37889300479327093436981769400, −3.25085509507549939780789668750, −3.20763896773857598840161580957, −3.18960377182551593238955110876, −2.58103571066133670247223067530, −2.14819968035149791608343776060, −1.73127445152277369859802967010, −1.33330068324424008595052916678, −0.28452335447041207470936144583, −0.22737874140136406708681448860,
0.22737874140136406708681448860, 0.28452335447041207470936144583, 1.33330068324424008595052916678, 1.73127445152277369859802967010, 2.14819968035149791608343776060, 2.58103571066133670247223067530, 3.18960377182551593238955110876, 3.20763896773857598840161580957, 3.25085509507549939780789668750, 3.37889300479327093436981769400, 3.75756267574416350282101829013, 4.11123116851441865629012377638, 4.23765173728633795794916638737, 4.41564703159337459781153452433, 4.58259226043927656285763485837, 4.74999179007308745763572296139, 5.11838058514613447582508660963, 5.11962945969051842312858822497, 5.32241656275636039305172397398, 5.66939374135429044815198702286, 5.73915444991662464568455798372, 6.11339804710674553480174302421, 6.51569285147248066793496773061, 6.55530885698656436411880909652, 6.65246216108314240589369818666
