| L(s) = 1 | − 16·2-s + 64·4-s + 1.02e3·8-s + 2.00e3·9-s − 3.16e3·11-s − 1.63e4·16-s − 3.20e4·18-s + 5.05e4·22-s − 2.00e5·23-s + 9.70e4·25-s + 4.63e4·29-s + 6.55e4·32-s + 1.28e5·36-s − 1.00e6·37-s − 2.24e6·43-s − 2.02e5·44-s + 3.20e6·46-s − 1.55e6·50-s − 2.57e6·53-s − 7.42e5·58-s + 7.86e5·64-s + 1.58e6·67-s − 8.91e6·71-s + 2.05e6·72-s + 1.60e7·74-s − 5.02e6·79-s + 4.78e6·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 0.707·8-s + 0.917·9-s − 0.715·11-s − 16-s − 1.29·18-s + 1.01·22-s − 3.43·23-s + 1.24·25-s + 0.353·29-s + 0.353·32-s + 0.458·36-s − 3.26·37-s − 4.30·43-s − 0.357·44-s + 4.85·46-s − 1.75·50-s − 2.37·53-s − 0.499·58-s + 3/8·64-s + 0.643·67-s − 2.95·71-s + 0.648·72-s + 4.61·74-s − 1.14·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0009869861452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0009869861452\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p^{3} T + p^{6} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 2006 T^{2} - 758933 T^{4} - 2006 p^{14} T^{6} + p^{28} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 3882 p^{2} T^{2} + 5304299 p^{4} T^{4} - 3882 p^{16} T^{6} + p^{28} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 1580 T - 16990771 T^{2} + 1580 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 27914122 T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 414811618 T^{2} + 3690851868376995 T^{4} - 414811618 p^{14} T^{6} + p^{28} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 1230606410 T^{2} + 715385450550203979 T^{4} + 1230606410 p^{14} T^{6} + p^{28} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 100152 T + 6625597657 T^{2} + 100152 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 11594 T + p^{7} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 50886712510 T^{2} + \)\(18\!\cdots\!79\)\( T^{4} - 50886712510 p^{14} T^{6} + p^{28} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 503058 T + 158135474231 T^{2} + 503058 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 189168160690 T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 560516 T + p^{7} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 927371942814 T^{2} + \)\(60\!\cdots\!27\)\( T^{4} - 927371942814 p^{14} T^{6} + p^{28} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 1287998 T + 484227708167 T^{2} + 1287998 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 1081866054630 T^{2} - \)\(50\!\cdots\!61\)\( T^{4} - 1081866054630 p^{14} T^{6} + p^{28} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 5968548381994 T^{2} + \)\(25\!\cdots\!95\)\( T^{4} - 5968548381994 p^{14} T^{6} + p^{28} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 792500 T - 5432655355323 T^{2} - 792500 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2229904 T + p^{7} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 14466065735054 T^{2} + \)\(87\!\cdots\!07\)\( T^{4} + 14466065735054 p^{14} T^{6} + p^{28} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 2513080 T - 12888337899759 T^{2} + 2513080 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 26023692819766 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 23586880683858 T^{2} - \)\(14\!\cdots\!77\)\( T^{4} - 23586880683858 p^{14} T^{6} + p^{28} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 78940465961026 T^{2} + p^{14} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.717974709427399592047562642210, −8.708238320354768714451704275381, −8.361420981990082869761797416423, −7.949172909332922364097945928445, −7.76900269833895795053291526454, −7.47064630867056358301676533999, −7.13779761702967345289537562729, −6.72857042810454033096815117698, −6.62406057140000830247485138044, −6.05344964255170918228997092607, −5.96176960434390864091394016959, −5.29343721854090048115055614081, −4.93492586712104281266058631060, −4.73799178811581948735492076221, −4.56912589864389099866828931782, −3.73492422173670529874935517375, −3.65021241076755231842150699131, −3.33275672248703618033709312993, −2.67998659725291350171479692793, −2.18559574608766681022695255385, −1.61061612021494771274371728038, −1.61056369410124416575660324663, −1.30505577496510803993286620669, −0.35645444864423423610875083509, −0.01137857701096321102806308205,
0.01137857701096321102806308205, 0.35645444864423423610875083509, 1.30505577496510803993286620669, 1.61056369410124416575660324663, 1.61061612021494771274371728038, 2.18559574608766681022695255385, 2.67998659725291350171479692793, 3.33275672248703618033709312993, 3.65021241076755231842150699131, 3.73492422173670529874935517375, 4.56912589864389099866828931782, 4.73799178811581948735492076221, 4.93492586712104281266058631060, 5.29343721854090048115055614081, 5.96176960434390864091394016959, 6.05344964255170918228997092607, 6.62406057140000830247485138044, 6.72857042810454033096815117698, 7.13779761702967345289537562729, 7.47064630867056358301676533999, 7.76900269833895795053291526454, 7.949172909332922364097945928445, 8.361420981990082869761797416423, 8.708238320354768714451704275381, 8.717974709427399592047562642210
