| L(s) = 1 | + 16·7-s − 4·9-s + 32·23-s + 16·25-s − 16·31-s − 32·47-s + 104·49-s − 64·63-s + 32·71-s + 16·73-s − 48·79-s + 10·81-s + 16·89-s − 32·97-s + 48·103-s − 16·113-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 512·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 6.04·7-s − 4/3·9-s + 6.67·23-s + 16/5·25-s − 2.87·31-s − 4.66·47-s + 14.8·49-s − 8.06·63-s + 3.79·71-s + 1.87·73-s − 5.40·79-s + 10/9·81-s + 1.69·89-s − 3.24·97-s + 4.72·103-s − 1.50·113-s + 5.09·121-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 + 40.3·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(31.40440721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(31.40440721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
| good | 5 | \( 1 - 16 T^{2} + 116 T^{4} - 112 p T^{6} + 2566 T^{8} - 112 p^{3} T^{10} + 116 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 8 T + 44 T^{2} - 24 p T^{3} + 510 T^{4} - 24 p^{2} T^{5} + 44 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 16 T^{2} + 452 T^{4} - 7856 T^{6} + 100006 T^{8} - 7856 p^{2} T^{10} + 452 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 36 T^{2} - 64 T^{3} + 614 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 88 T^{2} + 3772 T^{4} - 108456 T^{6} + 2345510 T^{8} - 108456 p^{2} T^{10} + 3772 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 29 | \( 1 - 144 T^{2} + 10036 T^{4} - 457904 T^{6} + 15291974 T^{8} - 457904 p^{2} T^{10} + 10036 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 8 T + 108 T^{2} + 616 T^{3} + 162 p T^{4} + 616 p T^{5} + 108 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 144 T^{2} + 9092 T^{4} - 335152 T^{6} + 10897638 T^{8} - 335152 p^{2} T^{10} + 9092 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 + 68 T^{2} - 64 T^{3} + 3206 T^{4} - 64 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 152 T^{2} + 13436 T^{4} - 859112 T^{6} + 42513894 T^{8} - 859112 p^{2} T^{10} + 13436 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 53 | \( 1 - 208 T^{2} + 22004 T^{4} - 1673840 T^{6} + 99995782 T^{8} - 1673840 p^{2} T^{10} + 22004 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 152 T^{2} + 17468 T^{4} - 1286632 T^{6} + 87350758 T^{8} - 1286632 p^{2} T^{10} + 17468 p^{4} T^{12} - 152 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 208 T^{2} + 15172 T^{4} - 338160 T^{6} - 4922074 T^{8} - 338160 p^{2} T^{10} + 15172 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 88 T^{2} + 12668 T^{4} - 867240 T^{6} + 78088998 T^{8} - 867240 p^{2} T^{10} + 12668 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 16 T + 4 p T^{2} - 2640 T^{3} + 28070 T^{4} - 2640 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 172 T^{2} - 1720 T^{3} + 16006 T^{4} - 1720 p T^{5} + 172 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 24 T + 396 T^{2} + 4344 T^{3} + 42398 T^{4} + 4344 p T^{5} + 396 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 504 T^{2} + 120700 T^{4} - 17869064 T^{6} + 1785000614 T^{8} - 17869064 p^{2} T^{10} + 120700 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 8 T + 188 T^{2} - 2424 T^{3} + 17894 T^{4} - 2424 p T^{5} + 188 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 356 T^{2} + 4400 T^{3} + 49990 T^{4} + 4400 p T^{5} + 356 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.50967678660311486799032623100, −3.42730274223565343834147856340, −3.23715443071158827920329535843, −3.18323738678047936114388647731, −3.16667747791961641456405065228, −3.12304240290548064519677151567, −2.93802948855013520095271016329, −2.77834836673844326691152587051, −2.76799530600685549694674607789, −2.47277603354276802653660681244, −2.25332352000691099150720963307, −2.18190007318647126490116933323, −2.17295713753005569606904762939, −1.91603407508617392232168752606, −1.78380110811524380991872173294, −1.57141648205243704570208329740, −1.53324533168771748075652642795, −1.53221452820273476656633193571, −1.33186019904760299780150906518, −1.25083482346848272024494521700, −1.08683781546487724886072393011, −0.78684478256853891510541897542, −0.66898198359746370396300153024, −0.65499976612598078142344493006, −0.20994494818707011428242107402,
0.20994494818707011428242107402, 0.65499976612598078142344493006, 0.66898198359746370396300153024, 0.78684478256853891510541897542, 1.08683781546487724886072393011, 1.25083482346848272024494521700, 1.33186019904760299780150906518, 1.53221452820273476656633193571, 1.53324533168771748075652642795, 1.57141648205243704570208329740, 1.78380110811524380991872173294, 1.91603407508617392232168752606, 2.17295713753005569606904762939, 2.18190007318647126490116933323, 2.25332352000691099150720963307, 2.47277603354276802653660681244, 2.76799530600685549694674607789, 2.77834836673844326691152587051, 2.93802948855013520095271016329, 3.12304240290548064519677151567, 3.16667747791961641456405065228, 3.18323738678047936114388647731, 3.23715443071158827920329535843, 3.42730274223565343834147856340, 3.50967678660311486799032623100
Plot not available for L-functions of degree greater than 10.
