L(s) = 1 | − 4·5-s + 16-s + 6·25-s + 4·37-s + 4·53-s − 4·73-s − 4·80-s + 8·97-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s − 16·185-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | − 4·5-s + 16-s + 6·25-s + 4·37-s + 4·53-s − 4·73-s − 4·80-s + 8·97-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + 181-s − 16·185-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{16} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4451083032\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4451083032\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T^{4} + T^{8} \) |
good | 3 | \( 1 - T^{8} + T^{16} \) |
| 5 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 11 | \( 1 - T^{8} + T^{16} \) |
| 13 | \( ( 1 + T^{2} )^{8} \) |
| 19 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 23 | \( 1 - T^{8} + T^{16} \) |
| 29 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 31 | \( 1 - T^{8} + T^{16} \) |
| 37 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 41 | \( ( 1 + T^{2} )^{4}( 1 + T^{4} )^{2} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 53 | \( ( 1 - T + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T^{2} + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 67 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 71 | \( ( 1 + T^{8} )^{2} \) |
| 73 | \( ( 1 + T + T^{2} )^{4}( 1 - T^{4} + T^{8} ) \) |
| 79 | \( 1 - T^{8} + T^{16} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T^{4} )^{2} \) |
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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
−3.70711713474621701676871875243, −3.63811592093273672741225142519, −3.60273123175848643784808390330, −3.54622319367956861352127656597, −3.49869172565451016974559020648, −3.46595150318003168243900735760, −3.27012524113220729940719904255, −2.86033546862184859570663045655, −2.81929553358394672626968431942, −2.68506171603975328910482062844, −2.64217750023780438875714466586, −2.46942171609113272626444025911, −2.44780871946282110045119261734, −2.33937338753010755157741205633, −2.33148123389208182931169255910, −1.97444567019157911734429425026, −1.79088260996609759252790880776, −1.51601818799201995453862550902, −1.46801307288539836521436001588, −1.24184580967821374918365488716, −1.09655978153054501608470272356, −1.05036628784670315363344964580, −0.789264978740373632525103578207, −0.52099088883558125701617172638, −0.30312208883794207120868664705,
0.30312208883794207120868664705, 0.52099088883558125701617172638, 0.789264978740373632525103578207, 1.05036628784670315363344964580, 1.09655978153054501608470272356, 1.24184580967821374918365488716, 1.46801307288539836521436001588, 1.51601818799201995453862550902, 1.79088260996609759252790880776, 1.97444567019157911734429425026, 2.33148123389208182931169255910, 2.33937338753010755157741205633, 2.44780871946282110045119261734, 2.46942171609113272626444025911, 2.64217750023780438875714466586, 2.68506171603975328910482062844, 2.81929553358394672626968431942, 2.86033546862184859570663045655, 3.27012524113220729940719904255, 3.46595150318003168243900735760, 3.49869172565451016974559020648, 3.54622319367956861352127656597, 3.60273123175848643784808390330, 3.63811592093273672741225142519, 3.70711713474621701676871875243
Plot not available for L-functions of degree greater than 10.