| L(s) = 1 | + 16·2-s + 192·4-s + 2.04e3·8-s − 2.00e3·9-s + 3.16e3·11-s + 2.04e4·16-s − 3.20e4·18-s + 5.05e4·22-s + 2.00e5·23-s − 9.70e4·25-s + 2.31e4·29-s + 1.96e5·32-s − 3.85e5·36-s + 1.00e6·37-s − 1.12e6·43-s + 6.06e5·44-s + 3.20e6·46-s − 1.55e6·50-s + 2.57e6·53-s + 3.71e5·58-s + 1.83e6·64-s − 1.58e6·67-s − 4.45e6·71-s − 4.10e6·72-s + 1.60e7·74-s + 5.02e6·79-s − 7.58e5·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.917·9-s + 0.715·11-s + 5/4·16-s − 1.29·18-s + 1.01·22-s + 3.43·23-s − 1.24·25-s + 0.176·29-s + 1.06·32-s − 1.37·36-s + 3.26·37-s − 2.15·43-s + 1.07·44-s + 4.85·46-s − 1.75·50-s + 2.37·53-s + 0.249·58-s + 7/8·64-s − 0.643·67-s − 1.47·71-s − 1.29·72-s + 4.61·74-s + 1.14·79-s − 0.158·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.136830931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.136830931\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2006 T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3882 p^{2} T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 1580 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 27914122 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 414811618 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 1230606410 T^{2} + p^{14} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 100152 T + p^{7} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 11594 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 50886712510 T^{2} + p^{14} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 503058 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 189168160690 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 560516 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 927371942814 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 1287998 T + p^{7} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1081866054630 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5968548381994 T^{2} + p^{14} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 792500 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2229904 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14466065735054 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2513080 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26023692819766 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 23586880683858 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 78940465961026 T^{2} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.12185761237938717465028871034, −12.20956159275434926888071403204, −11.65321926410270592527007978221, −11.53406318223773663447731387357, −10.90104105244933821213223886996, −10.37381619827247516136234072681, −9.394602363169766291181910513013, −9.113922423583384338503701268141, −8.223887044062306239288664283728, −7.66674236789323139349609238938, −6.70924255306165325775426096060, −6.64869417948291219924699464815, −5.58931035260215247166265868434, −5.36114651910294131884721608702, −4.46580920949440082519332042113, −3.93323256533135961231474666709, −2.89831596330642802283447527066, −2.77459749750194776341529133333, −1.51771043641590131424962381201, −0.72581451706167280933108389861,
0.72581451706167280933108389861, 1.51771043641590131424962381201, 2.77459749750194776341529133333, 2.89831596330642802283447527066, 3.93323256533135961231474666709, 4.46580920949440082519332042113, 5.36114651910294131884721608702, 5.58931035260215247166265868434, 6.64869417948291219924699464815, 6.70924255306165325775426096060, 7.66674236789323139349609238938, 8.223887044062306239288664283728, 9.113922423583384338503701268141, 9.394602363169766291181910513013, 10.37381619827247516136234072681, 10.90104105244933821213223886996, 11.53406318223773663447731387357, 11.65321926410270592527007978221, 12.20956159275434926888071403204, 13.12185761237938717465028871034
