| L(s) = 1 | − 2·2-s − 6·3-s + 2·4-s − 5·5-s + 12·6-s − 2·7-s − 5·8-s + 13·9-s + 10·10-s − 11-s − 12·12-s − 6·13-s + 4·14-s + 30·15-s + 5·16-s + 3·17-s − 26·18-s − 5·19-s − 10·20-s + 12·21-s + 2·22-s − 11·23-s + 30·24-s + 10·25-s + 12·26-s − 4·28-s − 10·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 3.46·3-s + 4-s − 2.23·5-s + 4.89·6-s − 0.755·7-s − 1.76·8-s + 13/3·9-s + 3.16·10-s − 0.301·11-s − 3.46·12-s − 1.66·13-s + 1.06·14-s + 7.74·15-s + 5/4·16-s + 0.727·17-s − 6.12·18-s − 1.14·19-s − 2.23·20-s + 2.61·21-s + 0.426·22-s − 2.29·23-s + 6.12·24-s + 2·25-s + 2.35·26-s − 0.755·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $D_{4}$ | \( ( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2:C_4$ | \( 1 + 2 T - 3 T^{2} + 10 T^{3} + 71 T^{4} + 10 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 3 T + 2 T^{2} - 75 T^{3} + 511 T^{4} - 75 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 11 T + 73 T^{2} + 475 T^{3} + 2796 T^{4} + 475 p T^{5} + 73 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 200 p T^{5} + 31 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 + 2 T + 33 T^{2} + 94 T^{3} + 1415 T^{4} + 94 p T^{5} + 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 3 T - 28 T^{2} + 195 T^{3} + 451 T^{4} + 195 p T^{5} - 28 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 7 T + 28 T^{2} + 389 T^{3} + 3975 T^{4} + 389 p T^{5} + 28 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 4 T + 43 T^{2} - 110 T^{3} + 741 T^{4} - 110 p T^{5} + 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 15 T + 31 T^{2} - 15 p T^{3} - 10424 T^{4} - 15 p^{2} T^{5} + 31 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 4 T + 106 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 12 T + 77 T^{2} + 1050 T^{3} + 13021 T^{4} + 1050 p T^{5} + 77 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 11 T + 48 T^{2} - 275 T^{3} - 6529 T^{4} - 275 p T^{5} + 48 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 25 T + 231 T^{2} + 1505 T^{3} + 12896 T^{4} + 1505 p T^{5} + 231 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 780 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 5 T + 61 T^{2} - 835 T^{3} + 12996 T^{4} - 835 p T^{5} + 61 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 33 T + 537 T^{2} - 6655 T^{3} + 71196 T^{4} - 6655 p T^{5} + 537 p^{2} T^{6} - 33 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.143783530935851392213755686712, −8.791756251539911373993450067624, −8.519619874491016171689475501119, −8.469656982779398906528946108719, −8.359959140364257831969711428085, −7.63691056127795391856522552241, −7.62533191381109672433060745955, −7.52068794829515440560939824633, −7.04455131950111521198953727258, −6.78383468243957799316965851725, −6.53107470300081332403285147583, −6.12468961722296087977295647840, −6.04648596813120984629333972874, −6.01160853646255828123391258481, −5.52156696148287760290434841296, −5.16030824847825101149257814584, −5.02089143230670445851977433704, −4.72971102953966555364102546576, −4.34155546405539228317852993214, −3.76992916348344598471175850878, −3.70106922447372372061430849958, −3.21288514454527297667830878118, −2.68393282191942426996149936751, −2.35375197216297440675671291878, −1.47776902871095647212477058488, 0, 0, 0, 0,
1.47776902871095647212477058488, 2.35375197216297440675671291878, 2.68393282191942426996149936751, 3.21288514454527297667830878118, 3.70106922447372372061430849958, 3.76992916348344598471175850878, 4.34155546405539228317852993214, 4.72971102953966555364102546576, 5.02089143230670445851977433704, 5.16030824847825101149257814584, 5.52156696148287760290434841296, 6.01160853646255828123391258481, 6.04648596813120984629333972874, 6.12468961722296087977295647840, 6.53107470300081332403285147583, 6.78383468243957799316965851725, 7.04455131950111521198953727258, 7.52068794829515440560939824633, 7.62533191381109672433060745955, 7.63691056127795391856522552241, 8.359959140364257831969711428085, 8.469656982779398906528946108719, 8.519619874491016171689475501119, 8.791756251539911373993450067624, 9.143783530935851392213755686712
