| L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s − 8·5-s + 16·6-s − 4·7-s − 12·8-s + 4·9-s + 32·10-s + 4·11-s − 32·12-s + 16·14-s + 32·15-s + 15·16-s + 8·17-s − 16·18-s − 8·19-s − 64·20-s + 16·21-s − 16·22-s − 8·23-s + 48·24-s + 40·25-s + 4·27-s − 32·28-s − 128·30-s − 12·31-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 4·4-s − 3.57·5-s + 6.53·6-s − 1.51·7-s − 4.24·8-s + 4/3·9-s + 10.1·10-s + 1.20·11-s − 9.23·12-s + 4.27·14-s + 8.26·15-s + 15/4·16-s + 1.94·17-s − 3.77·18-s − 1.83·19-s − 14.3·20-s + 3.49·21-s − 3.41·22-s − 1.66·23-s + 9.79·24-s + 8·25-s + 0.769·27-s − 6.04·28-s − 23.3·30-s − 2.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{3} T^{5} + p^{5} T^{6} + p^{5} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 4 T + 4 p T^{2} + 28 T^{3} + 56 T^{4} + 28 p T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 8 T + 24 T^{2} + 32 T^{3} + 32 T^{4} + 32 p T^{5} + 24 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 4 p T^{3} - 104 T^{4} - 4 p^{2} T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 170 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 96 T^{3} - 734 T^{4} - 96 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 54 T^{2} + 108 T^{3} + 162 T^{4} + 108 p T^{5} + 54 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 24 T^{2} - 368 T^{3} - 3168 T^{4} - 368 p T^{5} + 24 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 16 T + 114 T^{2} + 624 T^{3} + 3682 T^{4} + 624 p T^{5} + 114 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 6866 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 660 T^{3} + 6046 T^{4} - 660 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 332 T^{3} - 6386 T^{4} + 332 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T + 24 T^{2} - 32 T^{3} + 32 T^{4} - 32 p T^{5} + 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 84 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 4 T + 132 T^{2} + 580 T^{3} + 8984 T^{4} + 580 p T^{5} + 132 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 32 T + 456 T^{2} + 4264 T^{3} + 35616 T^{4} + 4264 p T^{5} + 456 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T + 242 T^{2} - 1632 T^{3} + 12002 T^{4} - 1632 p T^{5} + 242 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1584 T^{3} + 19346 T^{4} + 1584 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 11910 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 16 T + 82 T^{2} - 1648 T^{3} - 27614 T^{4} - 1648 p T^{5} + 82 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.84754813051447201204018049580, −7.68201996003824458037252210830, −7.67992152744237982596454702904, −7.27991930792491930005995485140, −6.90857399730255261500977813133, −6.89911745963119817772951027748, −6.88073174401892637654144615652, −6.39893166291528719343550040615, −6.14489389078576918637585420849, −6.13502362086243651298704364654, −5.66288876354892142787340630204, −5.35709781759557708417602410168, −5.28100550548386681609480212992, −5.01202075738896497441508316617, −4.54002746304117085643394825441, −3.90781461364585711173254420310, −3.86713323988229477778237157841, −3.85241020044506007706365457691, −3.72999970128369109803767404523, −3.21804298140366788022005569640, −2.71091861049034862228402428296, −2.67906135022428548548399029787, −1.80476335188985763961081751060, −1.26793441288875398764519362076, −0.982909405266385198144487805545, 0, 0, 0, 0,
0.982909405266385198144487805545, 1.26793441288875398764519362076, 1.80476335188985763961081751060, 2.67906135022428548548399029787, 2.71091861049034862228402428296, 3.21804298140366788022005569640, 3.72999970128369109803767404523, 3.85241020044506007706365457691, 3.86713323988229477778237157841, 3.90781461364585711173254420310, 4.54002746304117085643394825441, 5.01202075738896497441508316617, 5.28100550548386681609480212992, 5.35709781759557708417602410168, 5.66288876354892142787340630204, 6.13502362086243651298704364654, 6.14489389078576918637585420849, 6.39893166291528719343550040615, 6.88073174401892637654144615652, 6.89911745963119817772951027748, 6.90857399730255261500977813133, 7.27991930792491930005995485140, 7.67992152744237982596454702904, 7.68201996003824458037252210830, 7.84754813051447201204018049580
