| L(s) = 1 | + 4·7-s + 4·9-s + 16·17-s − 8·23-s + 12·25-s + 24·31-s + 8·47-s + 10·49-s + 16·63-s − 8·73-s + 32·79-s + 6·81-s + 8·89-s + 16·97-s + 8·103-s − 40·113-s + 64·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·153-s + 157-s − 32·161-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 4/3·9-s + 3.88·17-s − 1.66·23-s + 12/5·25-s + 4.31·31-s + 1.16·47-s + 10/7·49-s + 2.01·63-s − 0.936·73-s + 3.60·79-s + 2/3·81-s + 0.847·89-s + 1.62·97-s + 0.788·103-s − 3.76·113-s + 5.86·119-s + 1.09·121-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 + 5.17·153-s + 0.0798·157-s − 2.52·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.00374198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.00374198\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 714 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 330 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14954 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 7530 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 2826 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 19530 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 198 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.83431033255258583427547159594, −6.26411950007931699428408471275, −6.17561635763057983480054615596, −5.94336769717905165787924854229, −5.74739581938690429267788114005, −5.59474279966236127256845200622, −5.23744675204222577953929710174, −4.96324466409524794217469381680, −4.96319786421136577335121990107, −4.67628803967934502731106463455, −4.35512964513122782434486865249, −4.25348205614624460703389397183, −4.24277459031917661198440211684, −3.45573477805429780428993194472, −3.45451295076912313358052218485, −3.44733096056680486679184687144, −2.88062210475824555636089253444, −2.81386405086547800903504012379, −2.36499784299092570330532831133, −2.02091744065489837613140076739, −1.90299796008165212250908777058, −1.29514723183488182379261292073, −1.03469505783266015755837440082, −0.899703280123874887803585800784, −0.852737265068119918987929991369,
0.852737265068119918987929991369, 0.899703280123874887803585800784, 1.03469505783266015755837440082, 1.29514723183488182379261292073, 1.90299796008165212250908777058, 2.02091744065489837613140076739, 2.36499784299092570330532831133, 2.81386405086547800903504012379, 2.88062210475824555636089253444, 3.44733096056680486679184687144, 3.45451295076912313358052218485, 3.45573477805429780428993194472, 4.24277459031917661198440211684, 4.25348205614624460703389397183, 4.35512964513122782434486865249, 4.67628803967934502731106463455, 4.96319786421136577335121990107, 4.96324466409524794217469381680, 5.23744675204222577953929710174, 5.59474279966236127256845200622, 5.74739581938690429267788114005, 5.94336769717905165787924854229, 6.17561635763057983480054615596, 6.26411950007931699428408471275, 6.83431033255258583427547159594
