| L(s) = 1 | + 2-s − 3-s + 2·5-s − 6-s − 5·8-s + 2·9-s + 2·10-s − 11-s − 10·13-s − 2·15-s − 3·16-s − 5·17-s + 2·18-s − 6·19-s − 22-s + 2·23-s + 5·24-s + 25-s − 10·26-s + 27-s + 2·29-s − 2·30-s + 33-s − 5·34-s − 12·37-s − 6·38-s + 10·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.894·5-s − 0.408·6-s − 1.76·8-s + 2/3·9-s + 0.632·10-s − 0.301·11-s − 2.77·13-s − 0.516·15-s − 3/4·16-s − 1.21·17-s + 0.471·18-s − 1.37·19-s − 0.213·22-s + 0.417·23-s + 1.02·24-s + 1/5·25-s − 1.96·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s + 0.174·33-s − 0.857·34-s − 1.97·37-s − 0.973·38-s + 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7758230264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7758230264\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T + T^{2} + p^{2} T^{3} - 3 p T^{4} + p^{3} T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + T - T^{2} - 4 T^{3} - 8 T^{4} - 4 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + T - 17 T^{2} - 4 T^{3} + 192 T^{4} - 4 p T^{5} - 17 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 5 T - 11 T^{2} + 10 T^{3} + 582 T^{4} + 10 p T^{5} - 11 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 6 T^{2} - 48 T^{3} - 145 T^{4} - 48 p T^{5} + 6 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 26 T^{2} + 32 T^{3} + 279 T^{4} + 32 p T^{5} - 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 5 T - 37 T^{2} - 160 T^{3} + 460 T^{4} - 160 p T^{5} - 37 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2 T - 86 T^{2} + 32 T^{3} + 5079 T^{4} + 32 p T^{5} - 86 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T + 58 T^{2} + 864 T^{3} - 5433 T^{4} + 864 p T^{5} + 58 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T - 54 T^{2} - 256 T^{3} - 661 T^{4} - 256 p T^{5} - 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T - 30 T^{2} - 416 T^{3} - 1165 T^{4} - 416 p T^{5} - 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 9 T - 93 T^{2} - 144 T^{3} + 17636 T^{4} - 144 p T^{5} - 93 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 134 T^{2} + 48 T^{3} + 17775 T^{4} + 48 p T^{5} - 134 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.793444749671162391220199998741, −8.600717597746267950636138467893, −8.353016327153598469628050756959, −8.201652251255890066798978215209, −7.53255253917476033641248394724, −7.50821101639611709358043113070, −6.99202725755746316580776277279, −6.98141004779351009255926469604, −6.67190300903394685858393809605, −6.30365879538631817584928588462, −6.02907564875218583478125584766, −6.00454442166380771785770760453, −5.37999393960744924341863775734, −5.07019619584226018447242819914, −5.04609877550966369836172981249, −4.96285244183192653916253555799, −4.20676262882676258693889444437, −4.15370285427830873987576078279, −3.73497437938401043148341327549, −3.18903986882845799407296914638, −2.65464510224101773817650198727, −2.56092881061708253016277121846, −2.22004525095889461006466185654, −1.67573898910750981620620774125, −0.40771418801204113242815123513,
0.40771418801204113242815123513, 1.67573898910750981620620774125, 2.22004525095889461006466185654, 2.56092881061708253016277121846, 2.65464510224101773817650198727, 3.18903986882845799407296914638, 3.73497437938401043148341327549, 4.15370285427830873987576078279, 4.20676262882676258693889444437, 4.96285244183192653916253555799, 5.04609877550966369836172981249, 5.07019619584226018447242819914, 5.37999393960744924341863775734, 6.00454442166380771785770760453, 6.02907564875218583478125584766, 6.30365879538631817584928588462, 6.67190300903394685858393809605, 6.98141004779351009255926469604, 6.99202725755746316580776277279, 7.50821101639611709358043113070, 7.53255253917476033641248394724, 8.201652251255890066798978215209, 8.353016327153598469628050756959, 8.600717597746267950636138467893, 8.793444749671162391220199998741
