| L(s) = 1 | − 2·4-s − 6·11-s + 3·16-s − 6·19-s − 2·31-s + 2·41-s + 12·44-s + 3·49-s − 2·59-s + 18·61-s − 4·64-s − 6·71-s + 12·76-s + 10·79-s + 20·89-s − 14·101-s − 30·109-s − 121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s − 1.80·11-s + 3/4·16-s − 1.37·19-s − 0.359·31-s + 0.312·41-s + 1.80·44-s + 3/7·49-s − 0.260·59-s + 2.30·61-s − 1/2·64-s − 0.712·71-s + 1.37·76-s + 1.12·79-s + 2.11·89-s − 1.39·101-s − 2.87·109-s − 0.0909·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05720702914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05720702914\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 3 T^{2} + 8 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 35 T^{2} + 628 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 3 T + 30 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 123 T^{2} + 6428 T^{4} - 123 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 40 T^{2} - 2 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 5395 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + T + 108 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 93 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 134 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 5 T + 154 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 199 T^{2} + 21372 T^{4} - 199 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 144 T^{2} + 10718 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.49581999486699338154581736490, −5.38766298789659159898101835892, −5.37683820193734443211639999378, −5.17543946727932166329508234153, −5.05259643204581037110378371737, −4.72142197772678143095497659108, −4.58867183108736935619249509373, −4.26356039615854689577666172016, −4.22110443072731511381924566993, −3.97380703888279891457988498970, −3.92834629825321740830013087038, −3.54058451163298191438397200033, −3.24933086278471458165605094752, −3.18835277127837110534785680603, −3.06136439019000916426024322383, −2.47469667524298360828968110094, −2.38681239480376779599892567061, −2.27973483301547936810215092228, −2.25868507476286014158407198110, −1.75386455608993649491563654402, −1.37794086061822785142910042446, −1.13493562916460593838828628316, −0.909900953948142105451136225156, −0.41657384959657964327298468532, −0.04965785230790424375785183233,
0.04965785230790424375785183233, 0.41657384959657964327298468532, 0.909900953948142105451136225156, 1.13493562916460593838828628316, 1.37794086061822785142910042446, 1.75386455608993649491563654402, 2.25868507476286014158407198110, 2.27973483301547936810215092228, 2.38681239480376779599892567061, 2.47469667524298360828968110094, 3.06136439019000916426024322383, 3.18835277127837110534785680603, 3.24933086278471458165605094752, 3.54058451163298191438397200033, 3.92834629825321740830013087038, 3.97380703888279891457988498970, 4.22110443072731511381924566993, 4.26356039615854689577666172016, 4.58867183108736935619249509373, 4.72142197772678143095497659108, 5.05259643204581037110378371737, 5.17543946727932166329508234153, 5.37683820193734443211639999378, 5.38766298789659159898101835892, 5.49581999486699338154581736490
