| L(s) = 1 | + 12·4-s + 8·5-s + 79·16-s + 96·20-s + 36·23-s − 6·25-s − 80·31-s + 56·37-s − 16·47-s − 72·49-s − 140·53-s − 284·59-s + 360·64-s − 196·67-s − 4·71-s + 632·80-s − 336·89-s + 432·92-s + 112·97-s − 72·100-s − 124·103-s + 428·113-s + 288·115-s − 960·124-s − 192·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3·4-s + 8/5·5-s + 4.93·16-s + 24/5·20-s + 1.56·23-s − 0.239·25-s − 2.58·31-s + 1.51·37-s − 0.340·47-s − 1.46·49-s − 2.64·53-s − 4.81·59-s + 45/8·64-s − 2.92·67-s − 0.0563·71-s + 7.89·80-s − 3.77·89-s + 4.69·92-s + 1.15·97-s − 0.719·100-s − 1.20·103-s + 3.78·113-s + 2.50·115-s − 7.74·124-s − 1.53·125-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3307911048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3307911048\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 p^{2} T^{2} + 65 T^{4} - 3 p^{6} T^{6} + p^{8} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 4 T + 27 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 72 T^{2} + 3026 T^{4} + 72 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 232 T^{2} + 69255 T^{4} - 232 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 138039 T^{4} - 40 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1252 T^{2} + 645606 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 18 T + 776 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1080 T^{2} + 434759 T^{4} - 1080 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 40 T + 2022 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 28 T + 2787 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 1600 T^{2} + 4563279 T^{4} - 1600 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 160 p T^{2} + 18608994 T^{4} - 160 p^{5} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 966 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 70 T + 6411 T^{2} + 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 142 T + 10920 T^{2} + 142 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12072 T^{2} + 62535026 T^{4} - 12072 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 98 T + 10296 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 2 T + 7200 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 2760 T^{2} + 36243794 T^{4} + 2760 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 21316 T^{2} + 188444934 T^{4} - 21316 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 17124 T^{2} + 163904486 T^{4} - 17124 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 168 T + 21311 T^{2} + 168 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 56 T + 17079 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.04637115499782488251167127075, −6.34009513734194511648063722386, −6.28792683705608164688395567919, −6.28592505361340942611773858626, −6.09745908039917913191729971083, −5.84771683620337254757553966631, −5.84535759338333642616535808360, −5.34394734523080177040959428850, −5.06308056069894935396197078712, −5.02837919956169004585417874934, −4.55125055109742667482681102180, −4.27013769572892179156987509144, −4.20652051733565848351462487028, −3.40029446483691192393550627949, −3.31601914337248157848305019901, −3.15387096188920701337335765442, −2.99339157778196095541972346361, −2.64246848362855782807365103152, −2.39617545515612092647288261009, −1.97909617484023278833823403250, −1.69398571414790167437025757844, −1.62353034739046005726148586318, −1.50080423041066850889953268989, −1.06488172946944826791443733471, −0.04859158720820017295682216074,
0.04859158720820017295682216074, 1.06488172946944826791443733471, 1.50080423041066850889953268989, 1.62353034739046005726148586318, 1.69398571414790167437025757844, 1.97909617484023278833823403250, 2.39617545515612092647288261009, 2.64246848362855782807365103152, 2.99339157778196095541972346361, 3.15387096188920701337335765442, 3.31601914337248157848305019901, 3.40029446483691192393550627949, 4.20652051733565848351462487028, 4.27013769572892179156987509144, 4.55125055109742667482681102180, 5.02837919956169004585417874934, 5.06308056069894935396197078712, 5.34394734523080177040959428850, 5.84535759338333642616535808360, 5.84771683620337254757553966631, 6.09745908039917913191729971083, 6.28592505361340942611773858626, 6.28792683705608164688395567919, 6.34009513734194511648063722386, 7.04637115499782488251167127075
