| L(s) = 1 | − 8·3-s + 40·9-s − 152·27-s − 32·41-s + 64·43-s + 40·59-s + 48·73-s + 494·81-s − 64·83-s + 24·107-s − 32·113-s − 4·121-s + 256·123-s + 127-s − 512·129-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 320·177-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4.61·3-s + 40/3·9-s − 29.2·27-s − 4.99·41-s + 9.75·43-s + 5.20·59-s + 5.61·73-s + 54.8·81-s − 7.02·83-s + 2.32·107-s − 3.01·113-s − 0.363·121-s + 23.0·123-s + 0.0887·127-s − 45.0·129-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 − 24.0·177-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3200467958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3200467958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T^{8} \) |
| 17 | \( 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}( 1 - 8 T^{2} + 32 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 5 | \( 1 + p^{8} T^{16} \) |
| 7 | \( 1 + p^{8} T^{16} \) |
| 11 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}( 1 - 24 T^{2} + 288 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 13 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 24 T^{2} + 288 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + p^{8} T^{16} \) |
| 29 | \( 1 + p^{8} T^{16} \) |
| 31 | \( 1 + p^{8} T^{16} \) |
| 37 | \( 1 + p^{8} T^{16} \) |
| 41 | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2}( 1 - 96 T^{2} + 4608 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 43 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2}( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | \( 1 + p^{8} T^{16} \) |
| 67 | \( ( 1 + 168 T^{2} + 14112 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + p^{8} T^{16} \) |
| 73 | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2}( 1 + 48 T^{2} + 1152 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 79 | \( 1 + p^{8} T^{16} \) |
| 83 | \( ( 1 + 18 T + p T^{2} )^{4}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | \( ( 1 - 144 T^{2} + 10368 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )( 1 + 144 T^{2} + 10368 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 97 | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2}( 1 + 240 T^{2} + 28800 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} ) \) |
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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
−6.02614253883096055234443192276, −5.86526603978362069651694538888, −5.80196969712772629532827320002, −5.70811848678023755464304842518, −5.52485935603065002299893734233, −5.50184266676704327507019112591, −5.07137999219370845112006117957, −4.99443519482791595583765232423, −4.95320014526971389264820456042, −4.86018931396894971854764098565, −4.62570189285564866406207354960, −4.17816397323458371809694233499, −4.15046397426586435517510587630, −3.93709857503939816187447101185, −3.90328025965471079714129213129, −3.75109660258655357343784344030, −3.67703664544189989768355551706, −2.97117841450853895391762339108, −2.45304788197585007639762459122, −2.39567540208887780601475103499, −2.36875152739315199465400906459, −1.58856417793051259666983200814, −1.42791268339009044775333028186, −0.831865467626244818241506451007, −0.803130704621839629332318522429,
0.803130704621839629332318522429, 0.831865467626244818241506451007, 1.42791268339009044775333028186, 1.58856417793051259666983200814, 2.36875152739315199465400906459, 2.39567540208887780601475103499, 2.45304788197585007639762459122, 2.97117841450853895391762339108, 3.67703664544189989768355551706, 3.75109660258655357343784344030, 3.90328025965471079714129213129, 3.93709857503939816187447101185, 4.15046397426586435517510587630, 4.17816397323458371809694233499, 4.62570189285564866406207354960, 4.86018931396894971854764098565, 4.95320014526971389264820456042, 4.99443519482791595583765232423, 5.07137999219370845112006117957, 5.50184266676704327507019112591, 5.52485935603065002299893734233, 5.70811848678023755464304842518, 5.80196969712772629532827320002, 5.86526603978362069651694538888, 6.02614253883096055234443192276
Plot not available for L-functions of degree greater than 10.
