| L(s) = 1 | − 4·4-s − 8·9-s − 12·11-s + 7·16-s − 20·19-s + 4·29-s − 28·31-s + 32·36-s + 48·44-s − 16·49-s − 16·61-s − 8·64-s + 80·76-s + 4·79-s + 30·81-s + 96·99-s − 24·101-s + 16·109-s − 16·116-s + 52·121-s + 112·124-s + 127-s + 131-s + 137-s + 139-s − 56·144-s + 149-s + ⋯ |
| L(s) = 1 | − 2·4-s − 8/3·9-s − 3.61·11-s + 7/4·16-s − 4.58·19-s + 0.742·29-s − 5.02·31-s + 16/3·36-s + 7.23·44-s − 2.28·49-s − 2.04·61-s − 64-s + 9.17·76-s + 0.450·79-s + 10/3·81-s + 9.64·99-s − 2.38·101-s + 1.53·109-s − 1.48·116-s + 4.72·121-s + 10.0·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.66·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T^{2} + 9 T^{4} + p^{4} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 90 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 1266 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 14 T + 108 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 2034 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 88 T^{2} + 3906 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 7686 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 132 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 208 T^{2} + 22866 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 232 T^{2} + 30546 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.148755025073039547054745235999, −7.86193000717278034675113411491, −7.58567676377692858560630999224, −7.46722036521271238733943697362, −7.03685187772121780722721281515, −6.76781278717173684300163954268, −6.36249079994693833147146027653, −6.22935905249515518341454475017, −5.99493701983541844171006428405, −5.67747526733416859382256668644, −5.46644502432046420457460997002, −5.37607469323474750614680981479, −5.15965935230143700396427050862, −4.86865579449524575801560597716, −4.51830439893112707204826150168, −4.48121537717755380810610272301, −4.21261262011421621076784175938, −3.56665968211965972031565921894, −3.46674466468608182372265573095, −3.40386036389937436452446726420, −2.71424741700468938892273499279, −2.50297888709012953366744944613, −2.40910343119424981392956925862, −2.01203726423586468638355652595, −1.60144875518681194233903516435, 0, 0, 0, 0,
1.60144875518681194233903516435, 2.01203726423586468638355652595, 2.40910343119424981392956925862, 2.50297888709012953366744944613, 2.71424741700468938892273499279, 3.40386036389937436452446726420, 3.46674466468608182372265573095, 3.56665968211965972031565921894, 4.21261262011421621076784175938, 4.48121537717755380810610272301, 4.51830439893112707204826150168, 4.86865579449524575801560597716, 5.15965935230143700396427050862, 5.37607469323474750614680981479, 5.46644502432046420457460997002, 5.67747526733416859382256668644, 5.99493701983541844171006428405, 6.22935905249515518341454475017, 6.36249079994693833147146027653, 6.76781278717173684300163954268, 7.03685187772121780722721281515, 7.46722036521271238733943697362, 7.58567676377692858560630999224, 7.86193000717278034675113411491, 8.148755025073039547054745235999
