| L(s) = 1 | + 12·19-s − 4·31-s − 4·49-s + 28·61-s + 48·79-s + 56·109-s − 42·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | + 2.75·19-s − 0.718·31-s − 4/7·49-s + 3.58·61-s + 5.40·79-s + 5.36·109-s − 3.81·121-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 − 0.153·169-s + 0.0760·173-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 + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(26.51029385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(26.51029385\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 2 T^{2} - 30 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + T^{2} + 48 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 15 T^{2} + 602 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 24 T^{2} + 686 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 59 T^{2} + 3318 T^{4} + 59 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 129 T^{2} + 7232 T^{4} + 129 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 146 T^{2} + 8898 T^{4} + 146 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 18 T^{2} + 1274 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 134 T^{2} + 8946 T^{4} + 134 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 3 p T^{2} + 13988 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 7 T + 102 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 8 T^{2} - 3906 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 53 T^{2} + 3528 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 227 T^{2} + 22734 T^{4} + 227 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 + 210 T^{2} + 23642 T^{4} + 210 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 246 T^{2} + 28907 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 182 T^{2} + 16650 T^{4} + 182 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.33172480496491987866090571745, −2.93543177966222717100231466237, −2.89861087751840680271242942769, −2.85322905592995418293120666746, −2.69431074963731290549138721701, −2.64710091032351774019214831039, −2.63138939156651821211454046166, −2.57987890894357134364281087091, −2.26339060379925072646719808992, −2.03121265605403584115804780301, −1.93462420000399099692621897370, −1.93258998532918301217135056351, −1.79029508498385797437941119610, −1.69485324592483555795211047971, −1.67097452986605681025765262254, −1.66765527171259496989546726358, −1.51996453985927569394184835892, −0.919436241233365674612642531559, −0.889404841154574551545665243011, −0.835815594156000659035601418363, −0.800509856616437075921373256101, −0.72008104256037906846306939847, −0.56272909294713508404900535782, −0.50493238732193522933945870652, −0.18650309129911735215973950686,
0.18650309129911735215973950686, 0.50493238732193522933945870652, 0.56272909294713508404900535782, 0.72008104256037906846306939847, 0.800509856616437075921373256101, 0.835815594156000659035601418363, 0.889404841154574551545665243011, 0.919436241233365674612642531559, 1.51996453985927569394184835892, 1.66765527171259496989546726358, 1.67097452986605681025765262254, 1.69485324592483555795211047971, 1.79029508498385797437941119610, 1.93258998532918301217135056351, 1.93462420000399099692621897370, 2.03121265605403584115804780301, 2.26339060379925072646719808992, 2.57987890894357134364281087091, 2.63138939156651821211454046166, 2.64710091032351774019214831039, 2.69431074963731290549138721701, 2.85322905592995418293120666746, 2.89861087751840680271242942769, 2.93543177966222717100231466237, 3.33172480496491987866090571745
Plot not available for L-functions of degree greater than 10.
