| L(s) = 1 | + 8·4-s + 24·16-s − 8·19-s − 48·41-s − 4·49-s − 48·59-s − 24·61-s − 16·71-s − 64·76-s − 2·81-s + 24·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 384·164-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4·4-s + 6·16-s − 1.83·19-s − 7.49·41-s − 4/7·49-s − 6.24·59-s − 3.07·61-s − 1.89·71-s − 7.34·76-s − 2/9·81-s + 2.29·109-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 − 29.9·164-s + 0.0773·167-s + 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3291399315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3291399315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 13 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 + 2 T^{2} + 11 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 738 T^{4} + 260611 T^{8} - 738 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 4 T + 8 T^{2} - 92 T^{3} - 706 T^{4} - 92 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 82 T^{4} - 151437 T^{8} - 82 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 31 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 90 T^{2} + 3971 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 24 T + 288 T^{2} + 2448 T^{3} + 17087 T^{4} + 2448 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2}( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | \( ( 1 + 64 T^{2} + 2274 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 1598 T^{4} - 1400637 T^{8} + 1598 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 164 T^{2} + 14294 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 8 T + 32 T^{2} - 160 T^{3} - 7481 T^{4} - 160 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 208 T^{2} + 21426 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 12047 T^{4} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 362 T^{2} + 51491 T^{4} - 362 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.40751536378015578937536694933, −4.26057583502248076429575993198, −3.93389289515505025387269162016, −3.84320049625471115333413552002, −3.58372266970314975514213571921, −3.55142847862123418462709648409, −3.36024782776231887111903217202, −3.20994416659107175825842625686, −3.17044178120927572523927738524, −2.97648994488580056963403815144, −2.94313949999248358892698566527, −2.79177242598209592354363272080, −2.62727269809712788902464924981, −2.61263839225020512700691530282, −2.15941482382976557238570574292, −1.96659787742113864749354811769, −1.92492565319593449004144839771, −1.85138091249805560905234243434, −1.82497702744534381826520074396, −1.70288741715018417600935607709, −1.45890451130054961601956984256, −1.21415500593990526747619915748, −1.18643012438184014092550364318, −0.16820852232885409235583198051, −0.12386141711356014105886337824,
0.12386141711356014105886337824, 0.16820852232885409235583198051, 1.18643012438184014092550364318, 1.21415500593990526747619915748, 1.45890451130054961601956984256, 1.70288741715018417600935607709, 1.82497702744534381826520074396, 1.85138091249805560905234243434, 1.92492565319593449004144839771, 1.96659787742113864749354811769, 2.15941482382976557238570574292, 2.61263839225020512700691530282, 2.62727269809712788902464924981, 2.79177242598209592354363272080, 2.94313949999248358892698566527, 2.97648994488580056963403815144, 3.17044178120927572523927738524, 3.20994416659107175825842625686, 3.36024782776231887111903217202, 3.55142847862123418462709648409, 3.58372266970314975514213571921, 3.84320049625471115333413552002, 3.93389289515505025387269162016, 4.26057583502248076429575993198, 4.40751536378015578937536694933
Plot not available for L-functions of degree greater than 10.
