L(s) = 1 | − 8·2-s + 36·4-s − 120·8-s − 8·9-s + 330·16-s + 64·18-s + 8·23-s − 4·25-s − 792·32-s − 288·36-s − 64·46-s + 32·50-s − 40·53-s + 1.71e3·64-s + 960·72-s + 48·79-s + 30·81-s + 288·92-s − 144·100-s + 320·106-s − 32·109-s − 96·113-s + 44·121-s + 127-s − 3.43e3·128-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 5.65·2-s + 18·4-s − 42.4·8-s − 8/3·9-s + 82.5·16-s + 15.0·18-s + 1.66·23-s − 4/5·25-s − 140.·32-s − 48·36-s − 9.43·46-s + 4.52·50-s − 5.49·53-s + 214.5·64-s + 113.·72-s + 5.40·79-s + 10/3·81-s + 30.0·92-s − 14.3·100-s + 31.0·106-s − 3.06·109-s − 9.03·113-s + 4·121-s + 0.0887·127-s − 303.·128-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02182206551\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02182206551\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + T )^{8} \) |
| 3 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( 1 \) |
good | 11 | \( ( 1 - 2 p T^{2} + 243 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 6 T^{2} - 133 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 50 T^{2} + 1227 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 64 T^{2} + 2226 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 80 T^{2} + 3042 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 90 T^{2} + 4283 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 118 T^{2} + 6363 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 120 T^{2} + 6818 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2 T^{2} + 3339 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 5 T + p T^{2} )^{8} \) |
| 59 | \( ( 1 + 100 T^{2} + 7542 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 36 T^{2} + 1622 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 232 T^{2} + 23058 T^{4} - 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 136 T^{2} + 10962 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 12 T + 164 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 36 T^{2} - 3178 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 220 T^{2} + 26022 T^{4} + 220 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 232 T^{2} + 27954 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.89764317695389921583438192470, −3.87330205438505901124272572285, −3.66029344571901806020527654636, −3.43258787180919470095525158891, −3.27498190237387852867636385627, −3.25251039690017186892125516797, −3.03251288289690600165516106845, −2.82082542011733042052923415966, −2.81978062638585776760807031529, −2.75070532990502610782105288628, −2.62289294747781303989990975900, −2.50242728913624812642994384123, −2.44808239531683371584985846032, −2.08552017472706229988351810647, −1.96314945289196865808231244369, −1.74120294002365501975768635007, −1.64706582291894718329753453371, −1.61532086531485661076429697799, −1.47322095625135035917561249934, −1.15419089672038930816474269078, −0.896911659882268749178293639647, −0.880230480803108207190586077926, −0.51846805883156375540553485388, −0.30162534940911610021189028545, −0.10704242790980751892613349815,
0.10704242790980751892613349815, 0.30162534940911610021189028545, 0.51846805883156375540553485388, 0.880230480803108207190586077926, 0.896911659882268749178293639647, 1.15419089672038930816474269078, 1.47322095625135035917561249934, 1.61532086531485661076429697799, 1.64706582291894718329753453371, 1.74120294002365501975768635007, 1.96314945289196865808231244369, 2.08552017472706229988351810647, 2.44808239531683371584985846032, 2.50242728913624812642994384123, 2.62289294747781303989990975900, 2.75070532990502610782105288628, 2.81978062638585776760807031529, 2.82082542011733042052923415966, 3.03251288289690600165516106845, 3.25251039690017186892125516797, 3.27498190237387852867636385627, 3.43258787180919470095525158891, 3.66029344571901806020527654636, 3.87330205438505901124272572285, 3.89764317695389921583438192470
Plot not available for L-functions of degree greater than 10.