| L(s) = 1 | + 5-s + 6·9-s − 6·11-s + 2·19-s + 5·25-s − 2·29-s − 2·31-s − 30·41-s + 6·45-s − 6·55-s − 2·59-s − 24·61-s + 12·71-s − 14·79-s + 9·81-s + 28·89-s + 2·95-s − 36·99-s − 24·101-s + 16·109-s + 7·121-s + 14·125-s + 127-s + 131-s + 137-s + 139-s − 2·145-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 2·9-s − 1.80·11-s + 0.458·19-s + 25-s − 0.371·29-s − 0.359·31-s − 4.68·41-s + 0.894·45-s − 0.809·55-s − 0.260·59-s − 3.07·61-s + 1.42·71-s − 1.57·79-s + 81-s + 2.96·89-s + 0.205·95-s − 3.61·99-s − 2.38·101-s + 1.53·109-s + 7/11·121-s + 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.166·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4765907188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4765907188\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 956 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_{4}$ | \( ( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 17 T^{2} + 1656 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 7248 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 9032 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 9396 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + T + 104 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 62 T^{2} + 1731 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 25308 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 7 T + 156 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 28656 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.16299970214427679415899052316, −7.16053451874479466730082982188, −6.61569728695963710160373894604, −6.55189533247964646349836358030, −6.33452093959358487028270113513, −6.00934625557292535539049770195, −5.97895002338454314539707554362, −5.27761544728862968081407774533, −5.21747600458643885160863039927, −5.12262614404867285132776932416, −4.97758471321037013244516068047, −4.74155372741330535889366609832, −4.40872524139374407758221726370, −4.12936958292740178855440717374, −3.72775695241190780099359122473, −3.54953427983868203590556281641, −3.39068386288163115165096481636, −2.87246584958509110039018861677, −2.80284053534712485165337929509, −2.36985197177523433474671115118, −2.03410528293657867251889262524, −1.57719551285500936595430011296, −1.46707223415394666313660463887, −1.15338412196087249873991553514, −0.14982028937357912319505609570,
0.14982028937357912319505609570, 1.15338412196087249873991553514, 1.46707223415394666313660463887, 1.57719551285500936595430011296, 2.03410528293657867251889262524, 2.36985197177523433474671115118, 2.80284053534712485165337929509, 2.87246584958509110039018861677, 3.39068386288163115165096481636, 3.54953427983868203590556281641, 3.72775695241190780099359122473, 4.12936958292740178855440717374, 4.40872524139374407758221726370, 4.74155372741330535889366609832, 4.97758471321037013244516068047, 5.12262614404867285132776932416, 5.21747600458643885160863039927, 5.27761544728862968081407774533, 5.97895002338454314539707554362, 6.00934625557292535539049770195, 6.33452093959358487028270113513, 6.55189533247964646349836358030, 6.61569728695963710160373894604, 7.16053451874479466730082982188, 7.16299970214427679415899052316
