| L(s) = 1 | + 32·2-s − 162·3-s + 768·4-s + 1.02e3·5-s − 5.18e3·6-s − 1.62e3·7-s + 1.63e4·8-s + 1.96e4·9-s + 3.27e4·10-s + 1.32e4·11-s − 1.24e5·12-s − 5.71e4·13-s − 5.19e4·14-s − 1.65e5·15-s + 3.27e5·16-s + 3.38e5·17-s + 6.29e5·18-s + 2.35e5·19-s + 7.86e5·20-s + 2.63e5·21-s + 4.23e5·22-s + 8.55e5·23-s − 2.65e6·24-s − 1.83e6·25-s − 1.82e6·26-s − 2.12e6·27-s − 1.24e6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.732·5-s − 1.63·6-s − 0.255·7-s + 1.41·8-s + 9-s + 1.03·10-s + 0.272·11-s − 1.73·12-s − 0.554·13-s − 0.361·14-s − 0.846·15-s + 5/4·16-s + 0.984·17-s + 1.41·18-s + 0.415·19-s + 1.09·20-s + 0.295·21-s + 0.385·22-s + 0.637·23-s − 1.63·24-s − 0.942·25-s − 0.784·26-s − 0.769·27-s − 0.383·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.566029308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.566029308\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 1024 T + 115538 p^{2} T^{2} - 1024 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 232 p T + 380598 p^{2} T^{2} + 232 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 1204 p T + 4713668282 T^{2} - 1204 p^{10} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 338868 T + 205055574694 T^{2} - 338868 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 235944 T + 633846221678 T^{2} - 235944 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 855256 T + 2357859396926 T^{2} - 855256 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3026884 T + 29888279494526 T^{2} - 3026884 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4891136 T + 56246349038262 T^{2} - 4891136 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 18113340 T + 9140176667990 p T^{2} - 18113340 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 29481456 T + 775507746794506 T^{2} - 29481456 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 41647064 T + 1302540431931654 T^{2} - 41647064 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 52868756 T + 2916187899772994 T^{2} - 52868756 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10312404 T + 6621014285293966 T^{2} + 10312404 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17805940 T + 2529022349843162 T^{2} - 17805940 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 28581532 T + 22588181775855102 T^{2} + 28581532 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 69095008 T + 55577390850004734 T^{2} + 69095008 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 274598804 T + 85885889638557266 T^{2} - 274598804 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 43561364 T + 61426389477922134 T^{2} + 43561364 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5029744 T + 201308128592321886 T^{2} - 5029744 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 734192748 T + 455584875657096778 T^{2} - 734192748 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 50049432 T + 558925294891967674 T^{2} - 50049432 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 259995044 T + 1121212869704115942 T^{2} + 259995044 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−12.58734175003823927698427288973, −12.55002057590504410765201075497, −11.77746497643920114982613656565, −11.56693642986736264920201215080, −10.60190135427707953763239939477, −10.51546837907811391508412424269, −9.465647278944087632094402965580, −9.428233909963401999381429870848, −7.76892297414134667432654998633, −7.62794322798636290844432247486, −6.53063619038248655326748480816, −6.34453236146711585862518851100, −5.48049255275198194848198716161, −5.42108824196573620875507683758, −4.37824693150969376111274979529, −4.01927793644408247083506537420, −2.85152736924140867816952723417, −2.37254472621417969761745709325, −1.21528386150134086125633301394, −0.74985258267400563332626060450,
0.74985258267400563332626060450, 1.21528386150134086125633301394, 2.37254472621417969761745709325, 2.85152736924140867816952723417, 4.01927793644408247083506537420, 4.37824693150969376111274979529, 5.42108824196573620875507683758, 5.48049255275198194848198716161, 6.34453236146711585862518851100, 6.53063619038248655326748480816, 7.62794322798636290844432247486, 7.76892297414134667432654998633, 9.428233909963401999381429870848, 9.465647278944087632094402965580, 10.51546837907811391508412424269, 10.60190135427707953763239939477, 11.56693642986736264920201215080, 11.77746497643920114982613656565, 12.55002057590504410765201075497, 12.58734175003823927698427288973
