| L(s) = 1 | − 5·2-s + 4·3-s + 12·4-s + 2·5-s − 20·6-s + 7-s − 20·8-s + 3·9-s − 10·10-s + 48·12-s − 4·13-s − 5·14-s + 8·15-s + 30·16-s + 4·17-s − 15·18-s + 12·19-s + 24·20-s + 4·21-s − 8·23-s − 80·24-s + 5·25-s + 20·26-s − 10·27-s + 12·28-s + 14·29-s − 40·30-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 2.30·3-s + 6·4-s + 0.894·5-s − 8.16·6-s + 0.377·7-s − 7.07·8-s + 9-s − 3.16·10-s + 13.8·12-s − 1.10·13-s − 1.33·14-s + 2.06·15-s + 15/2·16-s + 0.970·17-s − 3.53·18-s + 2.75·19-s + 5.36·20-s + 0.872·21-s − 1.66·23-s − 16.3·24-s + 25-s + 3.92·26-s − 1.92·27-s + 2.26·28-s + 2.59·29-s − 7.30·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045607388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045607388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 25 p T^{5} + 13 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - 4 T + 13 T^{2} - 10 p T^{3} + 61 T^{4} - 10 p^{2} T^{5} + 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 12 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 4 T + 3 T^{2} + 50 T^{3} + 341 T^{4} + 50 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 4 T - T^{2} - 58 T^{3} + 509 T^{4} - 58 p T^{5} - p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 12 T + 45 T^{2} - 52 T^{3} + 9 T^{4} - 52 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 - 14 T + 47 T^{2} + 208 T^{3} - 2275 T^{4} + 208 p T^{5} + 47 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 10 T + 9 T^{2} + 10 p T^{3} - 2359 T^{4} + 10 p^{2} T^{5} + 9 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 14 T + 39 T^{2} + 272 T^{3} - 2611 T^{4} + 272 p T^{5} + 39 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 14 T + 55 T^{2} - 434 T^{3} - 5991 T^{4} - 434 p T^{5} + 55 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 7 T^{2} - 270 T^{3} + 1669 T^{4} - 270 p T^{5} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 14 T + 23 T^{2} - 400 T^{3} - 2899 T^{4} - 400 p T^{5} + 23 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 4 T - 43 T^{2} - 142 T^{3} + 4205 T^{4} - 142 p T^{5} - 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 21 T^{2} - 410 T^{3} + 2901 T^{4} - 410 p T^{5} - 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 4 T + 25 T^{2} + 596 T^{3} + 7569 T^{4} + 596 p T^{5} + 25 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 2 T - 49 T^{2} - 406 T^{3} + 5929 T^{4} - 406 p T^{5} - 49 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 20 T + 161 T^{2} - 1600 T^{3} + 18961 T^{4} - 1600 p T^{5} + 161 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 32 T + 461 T^{2} - 4596 T^{3} + 41849 T^{4} - 4596 p T^{5} + 461 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 26 T + 459 T^{2} - 6352 T^{3} + 72389 T^{4} - 6352 p T^{5} + 459 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.73519589734094867385253721065, −7.36395141035741389167215403531, −7.14683455222129430956120164306, −6.61236960017253930867164727758, −6.50616355943612034167987669765, −6.46233912785360185458527774648, −6.35213895544602688783696046210, −5.83488804446860787736878616851, −5.38332104307928491092898709794, −5.18157308281380591939021325858, −5.06340224851457316106961870832, −4.83624167887965794908445628535, −4.65784773074121400993553468477, −3.81542633798593752223528709530, −3.46560500483535225992482693794, −3.34663882314299484026709262457, −3.23651297372140709096408757499, −2.95768035574715319646249054185, −2.46910542114465016079972906756, −2.32065376511929835661828790468, −2.24692234904594493047724476117, −1.57371022906192222282030887624, −1.31189758006944710991201326104, −0.793923514844864063011695639305, −0.54420762736703976741782352982,
0.54420762736703976741782352982, 0.793923514844864063011695639305, 1.31189758006944710991201326104, 1.57371022906192222282030887624, 2.24692234904594493047724476117, 2.32065376511929835661828790468, 2.46910542114465016079972906756, 2.95768035574715319646249054185, 3.23651297372140709096408757499, 3.34663882314299484026709262457, 3.46560500483535225992482693794, 3.81542633798593752223528709530, 4.65784773074121400993553468477, 4.83624167887965794908445628535, 5.06340224851457316106961870832, 5.18157308281380591939021325858, 5.38332104307928491092898709794, 5.83488804446860787736878616851, 6.35213895544602688783696046210, 6.46233912785360185458527774648, 6.50616355943612034167987669765, 6.61236960017253930867164727758, 7.14683455222129430956120164306, 7.36395141035741389167215403531, 7.73519589734094867385253721065
