| L(s) = 1 | + 4·5-s − 16·13-s + 8·17-s + 14·25-s − 32·37-s − 8·41-s + 4·53-s + 12·61-s − 64·65-s − 8·73-s − 9·81-s + 32·85-s + 88·97-s + 52·101-s + 24·113-s − 32·121-s + 64·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 4.43·13-s + 1.94·17-s + 14/5·25-s − 5.26·37-s − 1.24·41-s + 0.549·53-s + 1.53·61-s − 7.93·65-s − 0.936·73-s − 81-s + 3.47·85-s + 8.93·97-s + 5.17·101-s + 2.25·113-s − 2.90·121-s + 5.72·125-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 + 9.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.909717208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.909717208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | \( 1 + 71 T^{4} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + p^{2} T^{4} )^{2}( 1 - p^{2} T^{4} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 + 16 T^{2} + 135 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 4 T + 8 T^{2} + 104 T^{3} - 497 T^{4} + 104 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 32 T^{2} + 663 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 697 T^{4} + 205968 T^{8} + 697 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 57 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | \( ( 1 + 16 T + 128 T^{2} + 864 T^{3} + 5543 T^{4} + 864 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 3577 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 2 T + 2 T^{2} + 208 T^{3} - 3017 T^{4} + 208 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 32 T^{2} - 2457 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 - 8183 T^{4} + 46810368 T^{8} - 8183 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 4 T + 8 T^{2} - 552 T^{3} - 6433 T^{4} - 552 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 136 T^{2} + 12255 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 5543 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 57 T^{2} - 4672 T^{4} + 57 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.87679727874017594130544567587, −4.72719741409895641870823050254, −4.61924539820769730787595521372, −4.51809075445508485699900255313, −3.96622674092309414154491441673, −3.94610996556922463831982860764, −3.90796955221812741119775232733, −3.76683698701370156428268347533, −3.43992685033603823902395658026, −3.28894059314305982375683549610, −3.22619547803167121673982748360, −3.05853000600049952973930822635, −3.01818238877325800940641768432, −2.84633774989759856270426917601, −2.53965562347722063144209943465, −2.21168146771897819336358809320, −2.14645897320885281058267528040, −2.11048668010361665783309347325, −2.09563622051096581504565418875, −1.77770178715128149176192498549, −1.47406202658271164358034222021, −1.46418875939908086889817860394, −0.800575135563416237115623434402, −0.71117556529027003996042627333, −0.29107762368289968905549254727,
0.29107762368289968905549254727, 0.71117556529027003996042627333, 0.800575135563416237115623434402, 1.46418875939908086889817860394, 1.47406202658271164358034222021, 1.77770178715128149176192498549, 2.09563622051096581504565418875, 2.11048668010361665783309347325, 2.14645897320885281058267528040, 2.21168146771897819336358809320, 2.53965562347722063144209943465, 2.84633774989759856270426917601, 3.01818238877325800940641768432, 3.05853000600049952973930822635, 3.22619547803167121673982748360, 3.28894059314305982375683549610, 3.43992685033603823902395658026, 3.76683698701370156428268347533, 3.90796955221812741119775232733, 3.94610996556922463831982860764, 3.96622674092309414154491441673, 4.51809075445508485699900255313, 4.61924539820769730787595521372, 4.72719741409895641870823050254, 4.87679727874017594130544567587
Plot not available for L-functions of degree greater than 10.
