| L(s) = 1 | − 8·2-s − 28·4-s + 456·8-s + 80·9-s − 246·16-s − 640·18-s − 124·23-s − 12·25-s + 312·29-s − 1.32e4·32-s − 2.24e3·36-s + 992·46-s − 384·49-s + 96·50-s − 2.49e3·58-s + 3.53e4·64-s + 3.44e3·71-s + 3.64e4·72-s + 1.08e3·81-s + 3.47e3·92-s + 3.07e3·98-s + 336·100-s − 8.73e3·116-s + 8.89e3·121-s + 127-s + 2.47e5·128-s + 131-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 7/2·4-s + 20.1·8-s + 2.96·9-s − 3.84·16-s − 8.38·18-s − 1.12·23-s − 0.0959·25-s + 1.99·29-s − 73.3·32-s − 10.3·36-s + 3.17·46-s − 1.11·49-s + 0.271·50-s − 5.65·58-s + 68.9·64-s + 5.76·71-s + 59.7·72-s + 1.48·81-s + 3.93·92-s + 3.16·98-s + 0.335·100-s − 6.99·116-s + 6.68·121-s + 0.000698·127-s + 171.·128-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.03492183946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03492183946\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 384 T^{2} + 2622 p^{2} T^{4} + 384 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | \( ( 1 + 62 T - 462 p T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| good | 2 | \( ( 1 + T + p^{3} T^{2} )^{8} \) |
| 3 | \( ( 1 - 20 T^{2} + p^{6} T^{4} )^{4} \) |
| 5 | \( ( 1 + 6 T^{2} - 4662 T^{4} + 6 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4446 T^{2} + 8448930 T^{4} - 4446 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 6658 T^{2} + 20699938 T^{4} - 6658 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 + 19158 T^{2} + 139996458 T^{4} + 19158 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2416 T^{2} + 54026350 T^{4} - 2416 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 78 T + 14378 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 - 9490 T^{2} + 1162115818 T^{4} - 9490 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 65094 T^{2} + 2111247978 T^{4} - 65094 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 - 70820 T^{2} + 795718 p^{2} T^{4} - 70820 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 - 174066 T^{2} + 15991399458 T^{4} - 174066 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 - 216130 T^{2} + 23335749258 T^{4} - 216130 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 7862 T^{2} + 41548048458 T^{4} - 7862 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 548584 T^{2} + 140975722146 T^{4} - 548584 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 + 701238 T^{2} + 220482793482 T^{4} + 701238 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 171438 T^{2} + 106252521858 T^{4} - 171438 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 - 862 T + 865662 T^{2} - 862 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 - 936220 T^{2} + 436605181078 T^{4} - 936220 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 38732 T^{2} - 6272373338 T^{4} - 38732 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 + 1567632 T^{2} + 1168547521518 T^{4} + 1567632 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 + 1330538 T^{2} + 896055032922 T^{4} + 1330538 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 + 1561626 T^{2} + 1464121316538 T^{4} + 1561626 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
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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
−5.21998034596156760689628180908, −4.91581648387531670988809106457, −4.90339043194285923310948533942, −4.80322304640327027858157602378, −4.77610722855011541814423209899, −4.50824449689643483457682610022, −4.30315103140294203612502694243, −4.24997533619553605580864432364, −4.15631520210194553220164136800, −4.15535146785484474550135286974, −3.65643574355506609155854116030, −3.61671226931219655261970650566, −3.46604410461892280218494657829, −3.38881974791101304656051919029, −3.15816782516290855060022183964, −2.31129844734837194104036036767, −1.96811695962049755669807488703, −1.93479918577275685230102708085, −1.34626299722782264405546678090, −1.22859235447362868784969298316, −1.13265437742751612884298248742, −1.01812641172274464070430000731, −0.63990628196875141226608962951, −0.44546337465277901974141247278, −0.10475180723656594614710010488,
0.10475180723656594614710010488, 0.44546337465277901974141247278, 0.63990628196875141226608962951, 1.01812641172274464070430000731, 1.13265437742751612884298248742, 1.22859235447362868784969298316, 1.34626299722782264405546678090, 1.93479918577275685230102708085, 1.96811695962049755669807488703, 2.31129844734837194104036036767, 3.15816782516290855060022183964, 3.38881974791101304656051919029, 3.46604410461892280218494657829, 3.61671226931219655261970650566, 3.65643574355506609155854116030, 4.15535146785484474550135286974, 4.15631520210194553220164136800, 4.24997533619553605580864432364, 4.30315103140294203612502694243, 4.50824449689643483457682610022, 4.77610722855011541814423209899, 4.80322304640327027858157602378, 4.90339043194285923310948533942, 4.91581648387531670988809106457, 5.21998034596156760689628180908
Plot not available for L-functions of degree greater than 10.
