| L(s) = 1 | + 702·2-s − 3.93e4·3-s + 1.08e5·4-s + 6.01e6·5-s − 2.76e7·6-s + 1.13e8·7-s + 1.74e8·8-s + 1.16e9·9-s + 4.22e9·10-s − 6.65e9·11-s − 4.26e9·12-s − 4.40e10·13-s + 7.99e10·14-s − 2.36e11·15-s + 6.41e10·16-s + 3.36e11·17-s + 8.15e11·18-s + 6.02e11·19-s + 6.51e11·20-s − 4.48e12·21-s − 4.66e12·22-s + 2.36e12·23-s − 6.85e12·24-s + 2.62e12·25-s − 3.09e13·26-s − 3.05e13·27-s + 1.23e13·28-s + ⋯ |
| L(s) = 1 | + 0.969·2-s − 1.15·3-s + 0.206·4-s + 1.37·5-s − 1.11·6-s + 1.06·7-s + 0.458·8-s + 9-s + 1.33·10-s − 0.850·11-s − 0.238·12-s − 1.15·13-s + 1.03·14-s − 1.59·15-s + 0.233·16-s + 0.687·17-s + 0.969·18-s + 0.428·19-s + 0.284·20-s − 1.23·21-s − 0.824·22-s + 0.274·23-s − 0.529·24-s + 0.137·25-s − 1.11·26-s − 0.769·27-s + 0.220·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(3.394121780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.394121780\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p^{9} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 351 p T + 6007 p^{6} T^{2} - 351 p^{20} T^{3} + p^{38} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 1203228 p T + 53712411262 p^{4} T^{2} - 1203228 p^{20} T^{3} + p^{38} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 113892064 T + 3260312560222818 p T^{2} - 113892064 p^{19} T^{3} + p^{38} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6650071272 T + 3471246282796372514 p T^{2} + 6650071272 p^{19} T^{3} + p^{38} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 44072356148 T + \)\(16\!\cdots\!42\)\( p T^{2} + 44072356148 p^{19} T^{3} + p^{38} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 19781263044 p T + \)\(82\!\cdots\!38\)\( p^{2} T^{2} - 19781263044 p^{20} T^{3} + p^{38} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 602118925096 T + \)\(31\!\cdots\!38\)\( T^{2} - 602118925096 p^{19} T^{3} + p^{38} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2368252165968 T + \)\(13\!\cdots\!94\)\( T^{2} - 2368252165968 p^{19} T^{3} + p^{38} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 280977251970492 T + \)\(31\!\cdots\!18\)\( T^{2} - 280977251970492 p^{19} T^{3} + p^{38} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1342262879152 p T + \)\(43\!\cdots\!42\)\( T^{2} - 1342262879152 p^{20} T^{3} + p^{38} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 637994163989884 T + \)\(27\!\cdots\!66\)\( T^{2} - 637994163989884 p^{19} T^{3} + p^{38} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1150038430280532 T + \)\(77\!\cdots\!62\)\( T^{2} - 1150038430280532 p^{19} T^{3} + p^{38} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2820518953538200 T + \)\(22\!\cdots\!90\)\( T^{2} - 2820518953538200 p^{19} T^{3} + p^{38} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5603091319500480 T + \)\(10\!\cdots\!90\)\( T^{2} + 5603091319500480 p^{19} T^{3} + p^{38} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 27537868857647628 T + \)\(13\!\cdots\!14\)\( T^{2} - 27537868857647628 p^{19} T^{3} + p^{38} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 55153474421835816 T + \)\(96\!\cdots\!18\)\( T^{2} + 55153474421835816 p^{19} T^{3} + p^{38} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 87192569066205620 T + \)\(18\!\cdots\!98\)\( T^{2} + 87192569066205620 p^{19} T^{3} + p^{38} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 499299710877878248 T + \)\(15\!\cdots\!82\)\( T^{2} - 499299710877878248 p^{19} T^{3} + p^{38} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 84915048436970544 T + \)\(22\!\cdots\!46\)\( T^{2} - 84915048436970544 p^{19} T^{3} + p^{38} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 241433711161728332 T + \)\(48\!\cdots\!74\)\( T^{2} + 241433711161728332 p^{19} T^{3} + p^{38} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1131591466033523440 T + \)\(99\!\cdots\!38\)\( T^{2} - 1131591466033523440 p^{19} T^{3} + p^{38} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1607743276003583784 T + \)\(43\!\cdots\!82\)\( T^{2} - 1607743276003583784 p^{19} T^{3} + p^{38} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4322797626369576276 T + \)\(13\!\cdots\!18\)\( T^{2} - 4322797626369576276 p^{19} T^{3} + p^{38} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7902201235299458308 T + \)\(12\!\cdots\!82\)\( T^{2} - 7902201235299458308 p^{19} T^{3} + p^{38} T^{4} \) |
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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
−21.70026776405241247383190315253, −21.67669866104074854585938684266, −20.91969252770191023602935180583, −19.50800368651085823387801775358, −17.90021638358707200352573178688, −17.84507865675774800695991832092, −16.85496353188924852403156201379, −15.79791731278784298424059453533, −14.36215004365440983920762826177, −13.83121466866768158656922427127, −12.79042370116953152185584785416, −11.86029359705413426139138588874, −10.60403474653141830489696772715, −9.845666150611256034314878520095, −7.75778586459758829259083930184, −6.25866995732094682835202364944, −5.03674457757287906743870906764, −4.82452618151653980702153629445, −2.36994551531524875620080423108, −1.03035876013938774134787752717,
1.03035876013938774134787752717, 2.36994551531524875620080423108, 4.82452618151653980702153629445, 5.03674457757287906743870906764, 6.25866995732094682835202364944, 7.75778586459758829259083930184, 9.845666150611256034314878520095, 10.60403474653141830489696772715, 11.86029359705413426139138588874, 12.79042370116953152185584785416, 13.83121466866768158656922427127, 14.36215004365440983920762826177, 15.79791731278784298424059453533, 16.85496353188924852403156201379, 17.84507865675774800695991832092, 17.90021638358707200352573178688, 19.50800368651085823387801775358, 20.91969252770191023602935180583, 21.67669866104074854585938684266, 21.70026776405241247383190315253
