| L(s) = 1 | + 31·4-s − 676·7-s − 3.44e3·13-s + 16-s − 3.66e3·19-s + 3.99e4·25-s − 2.09e4·28-s − 1.53e5·31-s + 1.28e5·37-s + 1.26e5·43-s + 5.41e4·49-s − 1.06e5·52-s + 1.67e5·61-s + 9.72e4·64-s + 8·67-s − 1.88e6·73-s − 1.13e5·76-s + 1.18e6·79-s + 2.33e6·91-s + 9.75e5·97-s + 1.23e6·100-s − 2.91e6·103-s + 1.79e6·109-s − 676·112-s + 2.37e5·121-s − 4.75e6·124-s + 127-s + ⋯ |
| L(s) = 1 | + 0.484·4-s − 1.97·7-s − 1.56·13-s + 0.000244·16-s − 0.534·19-s + 2.55·25-s − 0.954·28-s − 5.14·31-s + 2.54·37-s + 1.58·43-s + 0.460·49-s − 0.760·52-s + 0.738·61-s + 0.370·64-s + 2.65e−5·67-s − 4.83·73-s − 0.258·76-s + 2.41·79-s + 3.09·91-s + 1.06·97-s + 1.23·100-s − 2.66·103-s + 1.38·109-s − 0.000481·112-s + 0.133·121-s − 2.49·124-s + 1.05·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531441 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.05751107449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05751107449\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 15 p^{6} T^{4} - 31 p^{12} T^{6} + p^{24} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 39946 T^{2} + 1326939 p^{4} T^{4} - 39946 p^{12} T^{6} + p^{24} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 338 T + 144303 T^{2} + 338 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 237010 T^{2} + 2525538486003 T^{4} - 237010 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 1724 T + 9918438 T^{2} + 1724 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 89603068 T^{2} + 3171430277926278 T^{4} - 89603068 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 1832 T - 12669582 T^{2} + 1832 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 163687972 T^{2} + 40032274676179974 T^{4} - 163687972 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1305806620 T^{2} + 920788083150632358 T^{4} - 1305806620 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 76622 T + 3096283983 T^{2} + 76622 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 64480 T + 6063270018 T^{2} - 64480 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 16738605220 T^{2} + \)\(11\!\cdots\!78\)\( T^{4} - 16738605220 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 63004 T + 13577236998 T^{2} - 63004 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 38620471036 T^{2} + \)\(60\!\cdots\!30\)\( T^{4} - 38620471036 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 36815214586 T^{2} + \)\(92\!\cdots\!55\)\( T^{4} - 36815214586 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 32474880220 T^{2} + \)\(21\!\cdots\!78\)\( T^{4} - 32474880220 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 83848 T + 100016130498 T^{2} - 83848 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 35727479718 T^{2} - 4 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 40430249500 T^{2} - \)\(15\!\cdots\!82\)\( T^{4} - 40430249500 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 940838 T + 438394346043 T^{2} + 940838 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 594364 T + 526738516422 T^{2} - 594364 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 477023926306 T^{2} + \)\(10\!\cdots\!95\)\( T^{4} - 477023926306 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 300686549980 T^{2} + \)\(33\!\cdots\!18\)\( T^{4} - 300686549980 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 487606 T + 985315056603 T^{2} - 487606 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−11.68850162477862538934367008234, −11.23807923422349688427440998717, −10.94380486664315432848797359820, −10.82307789486735683815727368681, −10.29759850575312752134050842379, −9.778400741353619927688880912671, −9.712757088698401587196219388996, −9.169973564265543068494978770141, −8.924972782933606673051086079242, −8.774490473079452670235927196588, −7.74030963047841981072430607430, −7.51096003738820544113419519334, −7.22269592426998206064569776166, −6.84420842384134127255055409448, −6.42938884155899717089417501161, −5.94911344492520936529687517733, −5.56116539722861916486513866196, −4.98452367959994773267931509316, −4.42432489056320558551390835928, −3.71439030626939571547904683655, −3.28382371371266602177277263939, −2.61148876668330449667207642132, −2.30730205666125662886344636723, −1.21002099334173874007745561147, −0.06990772488473806550174332336,
0.06990772488473806550174332336, 1.21002099334173874007745561147, 2.30730205666125662886344636723, 2.61148876668330449667207642132, 3.28382371371266602177277263939, 3.71439030626939571547904683655, 4.42432489056320558551390835928, 4.98452367959994773267931509316, 5.56116539722861916486513866196, 5.94911344492520936529687517733, 6.42938884155899717089417501161, 6.84420842384134127255055409448, 7.22269592426998206064569776166, 7.51096003738820544113419519334, 7.74030963047841981072430607430, 8.774490473079452670235927196588, 8.924972782933606673051086079242, 9.169973564265543068494978770141, 9.712757088698401587196219388996, 9.778400741353619927688880912671, 10.29759850575312752134050842379, 10.82307789486735683815727368681, 10.94380486664315432848797359820, 11.23807923422349688427440998717, 11.68850162477862538934367008234
