| L(s) = 1 | − 2·3-s − 8·5-s − 12·7-s − 9-s − 8·11-s − 8·13-s + 16·15-s + 6·17-s − 8·19-s + 24·21-s − 14·23-s + 38·25-s + 2·27-s − 6·29-s + 16·33-s + 96·35-s − 12·37-s + 16·39-s + 20·41-s − 10·43-s + 8·45-s + 71·49-s − 12·51-s − 4·53-s + 64·55-s + 16·57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3.57·5-s − 4.53·7-s − 1/3·9-s − 2.41·11-s − 2.21·13-s + 4.13·15-s + 1.45·17-s − 1.83·19-s + 5.23·21-s − 2.91·23-s + 38/5·25-s + 0.384·27-s − 1.11·29-s + 2.78·33-s + 16.2·35-s − 1.97·37-s + 2.56·39-s + 3.12·41-s − 1.52·43-s + 1.19·45-s + 71/7·49-s − 1.68·51-s − 0.549·53-s + 8.62·55-s + 2.11·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 12 T + 73 T^{2} + 300 T^{3} + 912 T^{4} + 300 p T^{5} + 73 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 152 T^{3} + 532 T^{4} + 152 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 9 T^{2} + 6 p T^{3} - 688 T^{4} + 6 p^{2} T^{5} + 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 168 T^{3} + 644 T^{4} + 168 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 14 T + 53 T^{2} - 226 T^{3} - 2552 T^{4} - 226 p T^{5} + 53 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 1858 T^{4} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 12 T + 109 T^{2} + 732 T^{3} + 4128 T^{4} + 732 p T^{5} + 109 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 20 T + 125 T^{2} + 160 T^{3} - 5156 T^{4} + 160 p T^{5} + 125 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 29 T^{2} - 510 T^{3} - 5104 T^{4} - 510 p T^{5} + 29 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 2246 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 124 T^{3} + 1438 T^{4} + 124 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 65 T^{2} - 8 p T^{3} - 52 p T^{4} - 8 p^{2} T^{5} + 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 28 T + 457 T^{2} - 5404 T^{3} + 49912 T^{4} - 5404 p T^{5} + 457 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 688 T^{3} - 8612 T^{4} - 688 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 28614 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 21110 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 28 T + 197 T^{2} - 2408 T^{3} - 48788 T^{4} - 2408 p T^{5} + 197 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T - 83 T^{2} - 396 T^{3} + 28152 T^{4} - 396 p T^{5} - 83 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.413793070350809790817696357432, −8.896618256684059699562887867796, −8.631448819359183117117685580304, −8.515298786171705170278162602288, −8.067338261541585883227190416075, −7.71520440852332289957240188249, −7.61496261900778183029399949108, −7.56850270455174738563495620025, −7.36505334215355171349089517679, −6.72843737924971687083652450378, −6.63928451917899978981395186298, −6.40311872030447176096305503975, −6.33284359158995376358876461607, −5.66105279659693649096274484464, −5.40279072283639638831662990651, −5.34669825879048763921914900698, −5.09661681544614969419663511883, −4.20406615954836237219550413297, −4.08168167632930616719058901640, −3.97357922874094054930924309737, −3.68595093504021799051538127202, −3.23350443482501049199921909456, −2.80424179502574274612603317615, −2.73706155896064975018308445461, −2.47883410443463153285438854720, 0, 0, 0, 0,
2.47883410443463153285438854720, 2.73706155896064975018308445461, 2.80424179502574274612603317615, 3.23350443482501049199921909456, 3.68595093504021799051538127202, 3.97357922874094054930924309737, 4.08168167632930616719058901640, 4.20406615954836237219550413297, 5.09661681544614969419663511883, 5.34669825879048763921914900698, 5.40279072283639638831662990651, 5.66105279659693649096274484464, 6.33284359158995376358876461607, 6.40311872030447176096305503975, 6.63928451917899978981395186298, 6.72843737924971687083652450378, 7.36505334215355171349089517679, 7.56850270455174738563495620025, 7.61496261900778183029399949108, 7.71520440852332289957240188249, 8.067338261541585883227190416075, 8.515298786171705170278162602288, 8.631448819359183117117685580304, 8.896618256684059699562887867796, 9.413793070350809790817696357432
