| L(s) = 1 | + 2·2-s + 3-s + 4-s + 3·5-s + 2·6-s − 2·8-s + 3·9-s + 6·10-s − 3·11-s + 12-s + 4·13-s + 3·15-s − 4·16-s + 6·17-s + 6·18-s − 20·19-s + 3·20-s − 6·22-s + 9·23-s − 2·24-s + 4·25-s + 8·26-s + 8·27-s + 6·29-s + 6·30-s + 4·31-s − 2·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.816·6-s − 0.707·8-s + 9-s + 1.89·10-s − 0.904·11-s + 0.288·12-s + 1.10·13-s + 0.774·15-s − 16-s + 1.45·17-s + 1.41·18-s − 4.58·19-s + 0.670·20-s − 1.27·22-s + 1.87·23-s − 0.408·24-s + 4/5·25-s + 1.56·26-s + 1.53·27-s + 1.11·29-s + 1.09·30-s + 0.718·31-s − 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.259773515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.259773515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 720 p T^{5} + 95 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 3 T - 37 T^{2} - 216 T^{3} - 1896 T^{4} - 216 p T^{5} - 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 11 T + 43 T^{2} - 484 T^{3} - 5018 T^{4} - 484 p T^{5} + 43 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 1152 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - T - 119 T^{2} + 74 T^{3} + 4894 T^{4} + 74 p T^{5} - 119 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.36624592029444471228049046457, −6.65584274935182745376749030947, −6.61813138476618736180225682753, −6.48516152979091867566085934991, −6.48032582706566435465766279699, −6.08847781245883529924651428174, −5.79303530703679504942693763688, −5.70098194972540099873915646448, −5.57744561892566941631190842050, −5.00612751876087593455304802864, −4.75819663681857561496220244630, −4.71058777897684283939515800266, −4.39399906929075744339790282848, −4.21758354051815543219830372636, −4.17162404354524650813944691647, −3.64162141721823896375200343178, −3.37529735395568641732892505039, −2.97829142051715479256209401247, −2.65610113886163924124538008959, −2.57263949076170193493565583571, −2.54110089053881379362304815798, −1.79625300960837753439898942385, −1.39937089212765376867117038282, −1.32693635089189741031910185607, −0.51851934664225093502716716900,
0.51851934664225093502716716900, 1.32693635089189741031910185607, 1.39937089212765376867117038282, 1.79625300960837753439898942385, 2.54110089053881379362304815798, 2.57263949076170193493565583571, 2.65610113886163924124538008959, 2.97829142051715479256209401247, 3.37529735395568641732892505039, 3.64162141721823896375200343178, 4.17162404354524650813944691647, 4.21758354051815543219830372636, 4.39399906929075744339790282848, 4.71058777897684283939515800266, 4.75819663681857561496220244630, 5.00612751876087593455304802864, 5.57744561892566941631190842050, 5.70098194972540099873915646448, 5.79303530703679504942693763688, 6.08847781245883529924651428174, 6.48032582706566435465766279699, 6.48516152979091867566085934991, 6.61813138476618736180225682753, 6.65584274935182745376749030947, 7.36624592029444471228049046457
