| L(s) = 1 | − 4·5-s − 4·11-s − 12·17-s + 8·19-s − 4·23-s + 12·25-s − 8·37-s + 24·41-s − 8·47-s − 4·53-s + 16·55-s + 8·61-s + 8·71-s + 24·73-s − 16·79-s − 16·83-s + 48·85-s − 20·89-s − 32·95-s − 16·97-s − 4·101-s + 20·107-s − 16·109-s + 40·113-s + 16·115-s + 18·121-s − 40·125-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.20·11-s − 2.91·17-s + 1.83·19-s − 0.834·23-s + 12/5·25-s − 1.31·37-s + 3.74·41-s − 1.16·47-s − 0.549·53-s + 2.15·55-s + 1.02·61-s + 0.949·71-s + 2.80·73-s − 1.80·79-s − 1.75·83-s + 5.20·85-s − 2.11·89-s − 3.28·95-s − 1.62·97-s − 0.398·101-s + 1.93·107-s − 1.53·109-s + 3.76·113-s + 1.49·115-s + 1.63·121-s − 3.57·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9819045253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9819045253\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} + 8 T^{3} + 39 T^{4} + 8 p T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 16 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 76 T^{2} + 24 p T^{3} + 111 p T^{4} + 24 p^{2} T^{5} + 76 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} - 64 T^{3} + 539 T^{4} - 64 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 54 T^{2} + 1955 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 128 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T + 26 T^{2} - 448 T^{3} - 3773 T^{4} - 448 p T^{5} + 26 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 46 T^{2} - 1365 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 24 T^{2} + 272 T^{3} + 119 T^{4} + 272 p T^{5} - 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 6 T^{2} - 4453 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 304 T^{2} - 3024 T^{3} + 26607 T^{4} - 3024 p T^{5} + 304 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 16 T + 66 T^{2} + 512 T^{3} + 9635 T^{4} + 512 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 + 20 T + 140 T^{2} + 1640 T^{3} + 24079 T^{4} + 1640 p T^{5} + 140 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.11911756337946779151182357222, −5.81567683065275215170328676498, −5.64857954542030057072390917303, −5.51920180854712335262172116934, −5.14479552198942866134518828022, −5.06272989021683651987980597764, −4.63245330586814057794013645950, −4.62556562222593770973714086662, −4.57916155555809650427811964107, −4.23686415104960884194665184320, −4.05827579019996105667245671162, −3.79522021239092970325451849556, −3.64886489395511178843831043208, −3.41505769867040097208749679158, −3.06216272919840156267798276970, −2.87730695372829671418905648996, −2.75000495998640446432968927803, −2.52537602151510186099356528698, −2.10542937404522368420720231154, −1.91450692137660561945274335462, −1.81595359728974560910694134271, −1.04071961842699339710849860811, −0.998106392469401072542646527624, −0.46860577426901089374163852937, −0.25029649782409258807191860504,
0.25029649782409258807191860504, 0.46860577426901089374163852937, 0.998106392469401072542646527624, 1.04071961842699339710849860811, 1.81595359728974560910694134271, 1.91450692137660561945274335462, 2.10542937404522368420720231154, 2.52537602151510186099356528698, 2.75000495998640446432968927803, 2.87730695372829671418905648996, 3.06216272919840156267798276970, 3.41505769867040097208749679158, 3.64886489395511178843831043208, 3.79522021239092970325451849556, 4.05827579019996105667245671162, 4.23686415104960884194665184320, 4.57916155555809650427811964107, 4.62556562222593770973714086662, 4.63245330586814057794013645950, 5.06272989021683651987980597764, 5.14479552198942866134518828022, 5.51920180854712335262172116934, 5.64857954542030057072390917303, 5.81567683065275215170328676498, 6.11911756337946779151182357222
