| L(s) = 1 | − 4-s + 3·9-s + 2·11-s − 3·16-s + 12·19-s − 2·29-s − 3·36-s + 4·41-s − 2·44-s − 2·49-s + 16·59-s + 12·61-s + 3·64-s + 32·71-s − 12·76-s + 18·79-s − 7·81-s − 12·89-s + 6·99-s − 32·101-s − 46·109-s + 2·116-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 9-s + 0.603·11-s − 3/4·16-s + 2.75·19-s − 0.371·29-s − 1/2·36-s + 0.624·41-s − 0.301·44-s − 2/7·49-s + 2.08·59-s + 1.53·61-s + 3/8·64-s + 3.79·71-s − 1.37·76-s + 2.02·79-s − 7/9·81-s − 1.27·89-s + 0.603·99-s − 3.18·101-s − 4.40·109-s + 0.185·116-s − 3·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520613131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520613131\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 472 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 47 T^{2} + 1024 T^{4} - 47 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3934 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 99 T^{2} + 5912 T^{4} - 99 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 13294 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11734 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 10150 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 9 T + 174 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 135 T^{2} + 14768 T^{4} - 135 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−9.308538067901808379879984163159, −9.261338228670521472303535516259, −9.073878245378057383776968434764, −8.263573243497107765732447990748, −8.219272527875463299226035180402, −7.998732513706251784273081401710, −7.944618598253357275994836845905, −7.13642732478025094772087550949, −7.11259926744100988530058152192, −6.83270223698443783784000966318, −6.81917683160626994725379570752, −6.32854984995160221273028087113, −5.71978643248983676128468301785, −5.55326853687470932258968942019, −5.27504434173653871080047728415, −5.02314218023534529060404694834, −4.68008526645575399287219944177, −4.12956679267171323152191454580, −3.85566179114664969708336788933, −3.74986861588644646392379912603, −3.25111233793821000485396046763, −2.47410597584842334148276805115, −2.44097078858064708710288877194, −1.41353180269501736525083447880, −1.03512613646503030578168443128,
1.03512613646503030578168443128, 1.41353180269501736525083447880, 2.44097078858064708710288877194, 2.47410597584842334148276805115, 3.25111233793821000485396046763, 3.74986861588644646392379912603, 3.85566179114664969708336788933, 4.12956679267171323152191454580, 4.68008526645575399287219944177, 5.02314218023534529060404694834, 5.27504434173653871080047728415, 5.55326853687470932258968942019, 5.71978643248983676128468301785, 6.32854984995160221273028087113, 6.81917683160626994725379570752, 6.83270223698443783784000966318, 7.11259926744100988530058152192, 7.13642732478025094772087550949, 7.944618598253357275994836845905, 7.998732513706251784273081401710, 8.219272527875463299226035180402, 8.263573243497107765732447990748, 9.073878245378057383776968434764, 9.261338228670521472303535516259, 9.308538067901808379879984163159
