| L(s) = 1 | + 16·7-s − 2·16-s − 4·25-s − 16·31-s − 24·37-s + 128·49-s − 24·61-s − 8·67-s − 16·73-s − 24·97-s − 16·103-s − 32·112-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 64·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 6.04·7-s − 1/2·16-s − 4/5·25-s − 2.87·31-s − 3.94·37-s + 18.2·49-s − 3.07·61-s − 0.977·67-s − 1.87·73-s − 2.43·97-s − 1.57·103-s − 3.02·112-s + 8/11·121-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 + 0.0760·173-s − 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.503089659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503089659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 8 T + 32 T^{2} - 96 T^{3} + 263 T^{4} - 96 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4 T^{2} - 138 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 142 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 188 T^{4} - 45306 T^{8} + 188 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 32 T^{2} + 594 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 967 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 106 T^{2} + 4467 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 94 T^{2} + 5187 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 1778 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4249 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 4900 T^{4} + 13820838 T^{8} - 4900 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 16 T^{2} + 6162 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6 T + 125 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 4 T + 8 T^{2} + 168 T^{3} + 2903 T^{4} + 168 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 244 T^{2} + 24582 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 8 T + 32 T^{2} + 264 T^{3} + 578 T^{4} + 264 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 152 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 18478 T^{4} + 160575699 T^{8} + 18478 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 286 T^{2} + 35427 T^{4} + 286 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 12 T + 72 T^{2} + 804 T^{3} + 8078 T^{4} + 804 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.37217470949673743537607422707, −4.32160365000106473056249620900, −4.24789079529117757059060966011, −4.20944367428120121832196056923, −3.88467753336995343552816715053, −3.86824152042017350962699770587, −3.68179333287274667599204360361, −3.58308189630511486968999325132, −3.46802754471321195066033654746, −2.96421192600346751127961290941, −2.96348407372463565169171031142, −2.96323533047516036613928094588, −2.75597891464403113438217522839, −2.51892324939465283471098622763, −2.25953477930660575510024034361, −1.95357779836524057919737931128, −1.84930326946547421028839047507, −1.78308467067049358148133589090, −1.72698671857879689743399530451, −1.64151665450357136852260840944, −1.55292228730719105518381352780, −1.40033389860370606177566009212, −0.975611430122312595317942584442, −0.70002725093114265121194719885, −0.11206705172761905232282635660,
0.11206705172761905232282635660, 0.70002725093114265121194719885, 0.975611430122312595317942584442, 1.40033389860370606177566009212, 1.55292228730719105518381352780, 1.64151665450357136852260840944, 1.72698671857879689743399530451, 1.78308467067049358148133589090, 1.84930326946547421028839047507, 1.95357779836524057919737931128, 2.25953477930660575510024034361, 2.51892324939465283471098622763, 2.75597891464403113438217522839, 2.96323533047516036613928094588, 2.96348407372463565169171031142, 2.96421192600346751127961290941, 3.46802754471321195066033654746, 3.58308189630511486968999325132, 3.68179333287274667599204360361, 3.86824152042017350962699770587, 3.88467753336995343552816715053, 4.20944367428120121832196056923, 4.24789079529117757059060966011, 4.32160365000106473056249620900, 4.37217470949673743537607422707
Plot not available for L-functions of degree greater than 10.
