| L(s) = 1 | − 8·5-s − 4·9-s − 8·13-s + 24·25-s − 24·29-s − 8·37-s − 16·41-s + 32·45-s − 12·49-s − 24·53-s − 24·61-s + 64·65-s + 8·73-s + 2·81-s + 8·89-s + 32·97-s − 40·101-s + 24·109-s − 8·113-s + 32·117-s − 4·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + 192·145-s + ⋯ |
| L(s) = 1 | − 3.57·5-s − 4/3·9-s − 2.21·13-s + 24/5·25-s − 4.45·29-s − 1.31·37-s − 2.49·41-s + 4.77·45-s − 1.71·49-s − 3.29·53-s − 3.07·61-s + 7.93·65-s + 0.936·73-s + 2/9·81-s + 0.847·89-s + 3.24·97-s − 3.98·101-s + 2.29·109-s − 0.752·113-s + 2.95·117-s − 0.363·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 15.9·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + 4 T^{2} + 14 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 102 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 4 T^{2} + 238 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 36 T^{2} + 654 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 12 T^{2} + 1062 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 43 | $C_2^2:C_4$ | \( 1 + 164 T^{2} + 10414 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 140 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 228 T^{2} + 19950 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 108 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 164 T^{2} + 13390 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 76 T^{2} + 9958 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 132 T^{2} + 10446 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.63040951298216592383973846614, −7.52652449732715625744351533608, −7.29167480018454822019831693821, −6.96175893752256069957271933888, −6.95743542326345898783494149063, −6.33555295029445251777833414601, −6.29574250196488514096116216846, −6.18679384276719217130076174849, −5.59261332877385011858163286391, −5.45515415242771396900689349268, −5.21278838884424787142706455980, −5.01316937321092435341698336690, −4.80679132389222211891971027778, −4.44714224397315183569249620325, −4.41160023863796060801601914370, −3.92430224800147126112576187538, −3.78465618977271633163404782466, −3.48947464065825514257617612637, −3.30924941613294792746464540091, −3.14634779598684385349958264777, −3.06152846779798906572228357794, −2.30171965683242663102981795106, −2.06456757727294343422209624109, −1.80033280773671616323619610936, −1.36988685895243526404365371456, 0, 0, 0, 0,
1.36988685895243526404365371456, 1.80033280773671616323619610936, 2.06456757727294343422209624109, 2.30171965683242663102981795106, 3.06152846779798906572228357794, 3.14634779598684385349958264777, 3.30924941613294792746464540091, 3.48947464065825514257617612637, 3.78465618977271633163404782466, 3.92430224800147126112576187538, 4.41160023863796060801601914370, 4.44714224397315183569249620325, 4.80679132389222211891971027778, 5.01316937321092435341698336690, 5.21278838884424787142706455980, 5.45515415242771396900689349268, 5.59261332877385011858163286391, 6.18679384276719217130076174849, 6.29574250196488514096116216846, 6.33555295029445251777833414601, 6.95743542326345898783494149063, 6.96175893752256069957271933888, 7.29167480018454822019831693821, 7.52652449732715625744351533608, 7.63040951298216592383973846614
