| L(s) = 1 | − 8·7-s + 8·19-s + 4·25-s − 8·31-s + 8·37-s + 12·49-s − 8·103-s − 56·109-s + 28·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 1.83·19-s + 4/5·25-s − 1.43·31-s + 1.31·37-s + 12/7·49-s − 0.788·103-s − 5.36·109-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-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 + 2.46·169-s + 0.0760·173-s − 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.587239819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587239819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 2 T^{2} + 33 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 14 T^{2} + 129 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 32 T^{2} + 762 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 14 T^{2} - 351 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 8 T^{2} + 1050 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - T + p T^{2} )^{8} \) |
| 41 | \( ( 1 - 50 T^{2} + 3825 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 64 T^{2} + 3570 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 8 T^{2} - 1398 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4 T^{2} + 3030 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 128 T^{2} + 10410 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 136 T^{2} + 11010 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 266 T^{2} + 27753 T^{4} - 266 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 88 T^{2} + 2226 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 208 T^{2} + 22146 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 152 T^{2} + 13722 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 146 T^{2} + 13233 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 172 T^{2} + 21606 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.59491601991524254773666399466, −3.56610683286212394242794853068, −3.43513878904390566710168672460, −3.34970779435137782981383851708, −3.05147224486631224189786822523, −2.99540989080402118089023842689, −2.98699389986473856056529370554, −2.71836539686717933082298533608, −2.65888955610607990746564755312, −2.61428333783676743648707251509, −2.58715044915673088761415599183, −2.42297012342198197177690290857, −2.30879275957575001832926948158, −1.92020394732292205999879336740, −1.78134620773861925201347113320, −1.73874032363368046500095787202, −1.58154602854550319382700260484, −1.42606561025901687241758340471, −1.22847883729242143737330354093, −1.15199331182034683631104532574, −0.947912533406744085048649154120, −0.72943520556827056746071329179, −0.38546299892492679391648746798, −0.35090222149286493280818778992, −0.17700542817164241543237123332,
0.17700542817164241543237123332, 0.35090222149286493280818778992, 0.38546299892492679391648746798, 0.72943520556827056746071329179, 0.947912533406744085048649154120, 1.15199331182034683631104532574, 1.22847883729242143737330354093, 1.42606561025901687241758340471, 1.58154602854550319382700260484, 1.73874032363368046500095787202, 1.78134620773861925201347113320, 1.92020394732292205999879336740, 2.30879275957575001832926948158, 2.42297012342198197177690290857, 2.58715044915673088761415599183, 2.61428333783676743648707251509, 2.65888955610607990746564755312, 2.71836539686717933082298533608, 2.98699389986473856056529370554, 2.99540989080402118089023842689, 3.05147224486631224189786822523, 3.34970779435137782981383851708, 3.43513878904390566710168672460, 3.56610683286212394242794853068, 3.59491601991524254773666399466
Plot not available for L-functions of degree greater than 10.
