| L(s) = 1 | − 12·11-s + 16-s + 32·31-s − 84·41-s + 16·61-s + 9·81-s − 48·101-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 12·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 3.61·11-s + 1/4·16-s + 5.74·31-s − 13.1·41-s + 2.04·61-s + 81-s − 4.77·101-s + 3.09·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s − 0.904·176-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4351386390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4351386390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 23 T^{4} + p^{4} T^{8} )( 1 + 71 T^{4} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 142 T^{4} - 8397 T^{8} + 142 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 47 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 + 958 T^{4} + 637923 T^{8} + 958 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 46 T^{2} + 1275 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 21 T + 188 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 217 T^{4} - 3371712 T^{8} + 217 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 1054 T^{4} - 3768765 T^{8} + 1054 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 29 T^{2} - 2640 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 - 8183 T^{4} + 46810368 T^{8} - 8183 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 5617 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2}( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | \( 1 + 13294 T^{4} + 129272115 T^{8} + 13294 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 + 4657 T^{4} - 66841632 T^{8} + 4657 p^{4} T^{12} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.97245233580033949985592584252, −4.90123600954574312613076232472, −4.60076058180777853225636636410, −4.55854507460634459623752892171, −4.47151969675397148136500124785, −4.26040047818561177554415923432, −3.96715444969358188815098036970, −3.88487485221578646724998048494, −3.66892946758160359052545163007, −3.38991426959229186858895279610, −3.29546566246711301907190401366, −3.28212061155652647303128791698, −3.02359974528113291077145738441, −2.98281538743725725516000433511, −2.78971947913349366466335164194, −2.59253926475763213027421666137, −2.42620907496419391841846690218, −2.27761270376454924522887663157, −1.99133345595986607385407332683, −1.63326013519296556944833134473, −1.62370666241228614173726748757, −1.50536184200965566614119064797, −1.05130341901194963155795700792, −0.49338548179738134441149647277, −0.17976486837392910438750151891,
0.17976486837392910438750151891, 0.49338548179738134441149647277, 1.05130341901194963155795700792, 1.50536184200965566614119064797, 1.62370666241228614173726748757, 1.63326013519296556944833134473, 1.99133345595986607385407332683, 2.27761270376454924522887663157, 2.42620907496419391841846690218, 2.59253926475763213027421666137, 2.78971947913349366466335164194, 2.98281538743725725516000433511, 3.02359974528113291077145738441, 3.28212061155652647303128791698, 3.29546566246711301907190401366, 3.38991426959229186858895279610, 3.66892946758160359052545163007, 3.88487485221578646724998048494, 3.96715444969358188815098036970, 4.26040047818561177554415923432, 4.47151969675397148136500124785, 4.55854507460634459623752892171, 4.60076058180777853225636636410, 4.90123600954574312613076232472, 4.97245233580033949985592584252
Plot not available for L-functions of degree greater than 10.
