| L(s) = 1 | + 2·4-s + 16-s − 40·19-s − 8·31-s + 36·41-s − 8·49-s + 12·59-s − 32·61-s − 2·64-s − 48·71-s − 80·76-s + 8·79-s − 72·89-s + 24·101-s − 64·109-s − 4·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 72·164-s + 167-s + ⋯ |
| L(s) = 1 | + 4-s + 1/4·16-s − 9.17·19-s − 1.43·31-s + 5.62·41-s − 8/7·49-s + 1.56·59-s − 4.09·61-s − 1/4·64-s − 5.69·71-s − 9.17·76-s + 0.900·79-s − 7.63·89-s + 2.38·101-s − 6.13·109-s − 0.363·121-s − 1.43·124-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 + 5.62·164-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{24} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1664616035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1664616035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8 T^{2} + 46 T^{4} - 640 T^{6} - 5213 T^{8} - 640 p^{2} T^{10} + 46 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 + 32 T^{2} + 526 T^{4} + 5120 T^{6} + 46387 T^{8} + 5120 p^{2} T^{10} + 526 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 40 T^{2} + 1071 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 8 p T^{5} - 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T^{2} - 702 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 + 110 T^{2} + 139 p T^{4} + 266750 T^{6} + 11699428 T^{8} + 266750 p^{2} T^{10} + 139 p^{5} T^{12} + 110 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 68 T^{2} + 2506 T^{4} - 156400 T^{6} - 10608173 T^{8} - 156400 p^{2} T^{10} + 2506 p^{4} T^{12} + 68 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 128 T^{2} + 8850 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 6 T - 85 T^{2} - 18 T^{3} + 9036 T^{4} - 18 p T^{5} - 85 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 71 | \( ( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 4 T + 70 T^{2} + 848 T^{3} - 4589 T^{4} + 848 p T^{5} + 70 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 302 T^{2} + 54841 T^{4} + 6820670 T^{6} + 646505092 T^{8} + 6820670 p^{2} T^{10} + 54841 p^{4} T^{12} + 302 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 9 T + p T^{2} )^{8} \) |
| 97 | \( 1 + 2 p T^{2} + 9793 T^{4} + 18050 p T^{6} + 332586244 T^{8} + 18050 p^{3} T^{10} + 9793 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) |
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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
−4.05124396753137062726543635823, −4.04884570333463660245296891750, −3.99685387688611163999115370024, −3.93891803264593848234597031838, −3.68982792739994537230987592081, −3.59535885221106374483482912315, −3.08695356490953770915957004307, −2.99216698231820653674439213742, −2.87845288986357502187157676321, −2.80411736353517922731003428963, −2.71247221804013769164247670395, −2.65018979782470541525768875800, −2.64111953416455343705167776431, −2.21496490032630623058658811241, −2.15604683670352279201390564018, −2.00710926709487350284130116956, −1.91121521712688236787892436783, −1.88370365760185772119331460980, −1.51733003721263135997844622854, −1.43127492526251718525968555665, −1.29885019408487224300059686541, −1.14470413468480800921484901381, −0.47705313436321994142863397230, −0.22945485107743055707195482068, −0.10246353154306206935961105424,
0.10246353154306206935961105424, 0.22945485107743055707195482068, 0.47705313436321994142863397230, 1.14470413468480800921484901381, 1.29885019408487224300059686541, 1.43127492526251718525968555665, 1.51733003721263135997844622854, 1.88370365760185772119331460980, 1.91121521712688236787892436783, 2.00710926709487350284130116956, 2.15604683670352279201390564018, 2.21496490032630623058658811241, 2.64111953416455343705167776431, 2.65018979782470541525768875800, 2.71247221804013769164247670395, 2.80411736353517922731003428963, 2.87845288986357502187157676321, 2.99216698231820653674439213742, 3.08695356490953770915957004307, 3.59535885221106374483482912315, 3.68982792739994537230987592081, 3.93891803264593848234597031838, 3.99685387688611163999115370024, 4.04884570333463660245296891750, 4.05124396753137062726543635823
Plot not available for L-functions of degree greater than 10.
