| L(s) = 1 | + 43·4-s − 1.60e3·11-s + 245·16-s + 3.02e3·19-s + 2.60e3·29-s − 1.16e4·31-s − 800·41-s − 6.88e4·44-s + 3.50e4·49-s + 1.26e5·59-s − 9.82e4·61-s − 1.44e4·64-s − 1.30e5·71-s + 1.30e5·76-s + 9.25e4·79-s − 1.74e5·89-s + 5.33e5·101-s + 2.53e4·109-s + 1.11e5·116-s + 9.84e5·121-s − 5.00e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.34·4-s − 3.98·11-s + 0.239·16-s + 1.92·19-s + 0.574·29-s − 2.17·31-s − 0.0743·41-s − 5.35·44-s + 2.08·49-s + 4.72·59-s − 3.38·61-s − 0.439·64-s − 3.06·71-s + 2.58·76-s + 1.66·79-s − 2.32·89-s + 5.20·101-s + 0.204·109-s + 0.771·116-s + 6.11·121-s − 2.92·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8678972362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8678972362\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 43 T^{2} + 401 p^{2} T^{4} - 43 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 5004 p T^{2} + 778890694 T^{4} - 5004 p^{11} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 800 T + 467602 T^{2} + 800 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1013972 T^{2} + 529410987894 T^{4} - 1013972 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3723388 T^{2} + 7358187518534 T^{4} - 3723388 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 1512 T + 1811734 T^{2} - 1512 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6458012 T^{2} + 77235343583334 T^{4} - 6458012 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1300 T - 17947202 T^{2} - 1300 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5824 T + 58720046 T^{2} + 5824 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 48163028 T^{2} - 1660934543432106 T^{4} - 48163028 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 400 T + 164704402 T^{2} + 400 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 139518572 T^{2} + 8922531748998294 T^{4} - 139518572 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 735942268 T^{2} + 240161694925514054 T^{4} - 735942268 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 422835532 T^{2} + 361691322258144854 T^{4} - 422835532 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 63200 T + 2393594098 T^{2} - 63200 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 49116 T + 1219519966 T^{2} + 49116 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 601773228 T^{2} + 3647854157687107894 T^{4} - 601773228 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 65200 T + 4591816702 T^{2} + 65200 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1954836572 T^{2} + 2094894445009493094 T^{4} - 1954836572 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 46288 T + 6429395534 T^{2} - 46288 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 5767053932 T^{2} + 25620183348039240054 T^{4} - 5767053932 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 87000 T + 10502568898 T^{2} + 87000 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 33696200828 T^{2} + \)\(43\!\cdots\!94\)\( T^{4} - 33696200828 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.73426330393971488638575889279, −7.69026223739498864835190988791, −7.59521119806531537116458811046, −7.33821837286885973956636114332, −7.03764370100428533059520672121, −6.76548521343513946765976697339, −6.57291434398747848316683508886, −5.87641473139262924516175822873, −5.61942219278532769847048665499, −5.55041771005853435957304894792, −5.53677651745704887496722919483, −5.06029622003251896044872287391, −4.71809832573273986249583563676, −4.43210607409944372839456060104, −4.01648772792433214573753010684, −3.33863871414014950549384100830, −3.24222856645599720151099909122, −2.92242862041399010689416299652, −2.65013969616019857167769660951, −2.25233136588045528601240215463, −2.02980610074143077984414373723, −1.76911981011025841484399759299, −0.931027630683367652407757591121, −0.71065870457587755136485036342, −0.13212078541328388724838597065,
0.13212078541328388724838597065, 0.71065870457587755136485036342, 0.931027630683367652407757591121, 1.76911981011025841484399759299, 2.02980610074143077984414373723, 2.25233136588045528601240215463, 2.65013969616019857167769660951, 2.92242862041399010689416299652, 3.24222856645599720151099909122, 3.33863871414014950549384100830, 4.01648772792433214573753010684, 4.43210607409944372839456060104, 4.71809832573273986249583563676, 5.06029622003251896044872287391, 5.53677651745704887496722919483, 5.55041771005853435957304894792, 5.61942219278532769847048665499, 5.87641473139262924516175822873, 6.57291434398747848316683508886, 6.76548521343513946765976697339, 7.03764370100428533059520672121, 7.33821837286885973956636114332, 7.59521119806531537116458811046, 7.69026223739498864835190988791, 7.73426330393971488638575889279
