| L(s) = 1 | − 2-s − 2·3-s − 2·4-s − 19·5-s + 2·6-s + 40·7-s + 23·8-s + 55·9-s + 19·10-s − 66·11-s + 4·12-s − 101·13-s − 40·14-s + 38·15-s + 33·16-s + 75·17-s − 55·18-s + 57·19-s + 38·20-s − 80·21-s + 66·22-s − 23-s − 46·24-s + 322·25-s + 101·26-s − 214·27-s − 80·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.384·3-s − 1/4·4-s − 1.69·5-s + 0.136·6-s + 2.15·7-s + 1.01·8-s + 2.03·9-s + 0.600·10-s − 1.80·11-s + 0.0962·12-s − 2.15·13-s − 0.763·14-s + 0.654·15-s + 0.515·16-s + 1.07·17-s − 0.720·18-s + 0.688·19-s + 0.424·20-s − 0.831·21-s + 0.639·22-s − 0.00906·23-s − 0.391·24-s + 2.57·25-s + 0.761·26-s − 1.52·27-s − 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130321 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130321 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8386838497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8386838497\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 19 | $C_2^2$ | \( 1 - 3 p T + 22 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + T + 3 T^{2} - 9 p T^{3} - 17 p^{2} T^{4} - 9 p^{4} T^{5} + 3 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + T - 26 T^{2} + p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 + 19 T + 39 T^{2} + 1368 T^{3} + 42934 T^{4} + 1368 p^{3} T^{5} + 39 p^{6} T^{6} + 19 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 20 T + 494 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 p T + 2770 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 101 T + 3713 T^{2} + 211494 T^{3} + 14855738 T^{4} + 211494 p^{3} T^{5} + 3713 p^{6} T^{6} + 101 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 75 T + 2441 T^{2} + 498150 T^{3} - 41635338 T^{4} + 498150 p^{3} T^{5} + 2441 p^{6} T^{6} - 75 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + T - 19059 T^{2} - 5274 T^{3} + 215235544 T^{4} - 5274 p^{3} T^{5} - 19059 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 85 T - 12681 T^{2} + 2454120 T^{3} - 374785010 T^{4} + 2454120 p^{3} T^{5} - 12681 p^{6} T^{6} - 85 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 22 T + 59046 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 448 T + 127830 T^{2} - 448 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 124 T - 126237 T^{2} + 467604 T^{3} + 14244408232 T^{4} + 467604 p^{3} T^{5} - 126237 p^{6} T^{6} + 124 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 311 T - 78425 T^{2} - 5017052 T^{3} + 16664761720 T^{4} - 5017052 p^{3} T^{5} - 78425 p^{6} T^{6} - 311 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 411 T - 53197 T^{2} + 5947992 T^{3} + 21019305612 T^{4} + 5947992 p^{3} T^{5} - 53197 p^{6} T^{6} + 411 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 261 T - 67795 T^{2} - 42239718 T^{3} - 13832852190 T^{4} - 42239718 p^{3} T^{5} - 67795 p^{6} T^{6} + 261 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 204 T - 189673 T^{2} - 36611676 T^{3} + 2767015416 T^{4} - 36611676 p^{3} T^{5} - 189673 p^{6} T^{6} + 204 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 531 T - 241597 T^{2} + 36955476 T^{3} + 158592815262 T^{4} + 36955476 p^{3} T^{5} - 241597 p^{6} T^{6} + 531 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 556 T + 109279 T^{2} - 223327964 T^{3} - 143492232488 T^{4} - 223327964 p^{3} T^{5} + 109279 p^{6} T^{6} + 556 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 1563 T + 1116569 T^{2} + 954333414 T^{3} + 756871198320 T^{4} + 954333414 p^{3} T^{5} + 1116569 p^{6} T^{6} + 1563 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 234 T - 595639 T^{2} + 29867526 T^{3} + 250378414884 T^{4} + 29867526 p^{3} T^{5} - 595639 p^{6} T^{6} - 234 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 331 T - 886369 T^{2} - 3261012 T^{3} + 694771263500 T^{4} - 3261012 p^{3} T^{5} - 886369 p^{6} T^{6} - 331 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1459 T + 1202686 T^{2} - 1459 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 601 T - 1025139 T^{2} - 14182398 T^{3} + 1170321796870 T^{4} - 14182398 p^{3} T^{5} - 1025139 p^{6} T^{6} + 601 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 324 T - 376477 T^{2} + 435421332 T^{3} - 696983774568 T^{4} + 435421332 p^{3} T^{5} - 376477 p^{6} T^{6} - 324 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−13.38825612845085032638649998888, −13.24555098785439847644393943663, −12.73219504325400140510692688207, −12.58165793606914439086535108391, −11.99535748711843374482209828702, −11.94719232392901472865016607469, −11.39059954791368989518783844836, −10.95287816099280060594172338651, −10.81353709486644099513906672459, −10.17650002544382095244424616725, −10.04640093347099595472234895511, −9.673628074594725531367843439653, −9.090306010611435308001886336950, −8.040001423454698361626527504655, −7.966303031422739308937698687555, −7.65917478295079727589729720529, −7.60484333006746808795575199932, −7.37076504360630409060805599513, −6.33350131973270507163030434249, −5.21346500966785888043123614810, −4.93989408625555646310977864210, −4.52980430317438357480991925445, −4.33648410328182797990327425140, −2.88838182797516627647302683780, −1.28359854954818164315969320333,
1.28359854954818164315969320333, 2.88838182797516627647302683780, 4.33648410328182797990327425140, 4.52980430317438357480991925445, 4.93989408625555646310977864210, 5.21346500966785888043123614810, 6.33350131973270507163030434249, 7.37076504360630409060805599513, 7.60484333006746808795575199932, 7.65917478295079727589729720529, 7.966303031422739308937698687555, 8.040001423454698361626527504655, 9.090306010611435308001886336950, 9.673628074594725531367843439653, 10.04640093347099595472234895511, 10.17650002544382095244424616725, 10.81353709486644099513906672459, 10.95287816099280060594172338651, 11.39059954791368989518783844836, 11.94719232392901472865016607469, 11.99535748711843374482209828702, 12.58165793606914439086535108391, 12.73219504325400140510692688207, 13.24555098785439847644393943663, 13.38825612845085032638649998888
