| L(s) = 1 | + 2·5-s − 9-s + 2·17-s + 3·25-s + 2·37-s − 2·45-s + 2·53-s − 2·61-s − 2·73-s + 81-s + 4·85-s − 2·89-s + 2·101-s − 2·109-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s − 4·169-s + ⋯ |
| L(s) = 1 | + 2·5-s − 9-s + 2·17-s + 3·25-s + 2·37-s − 2·45-s + 2·53-s − 2·61-s − 2·73-s + 81-s + 4·85-s − 2·89-s + 2·101-s − 2·109-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + 163-s + 167-s − 4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.930576185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930576185\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.91706419275973185287952284097, −6.51138610755418176410632535995, −6.40967540526245773432452434970, −6.23394325079814120440714645491, −6.09424971939573400825935975273, −5.81867498413783790780442073604, −5.50792528779483325711728324327, −5.50093497655651430387291733205, −5.49871722901946486139459661329, −4.96068044842285934995907540495, −4.90738215615066665557081244035, −4.49873285644506070840628310648, −4.39176794211603451297949411961, −4.05779209180418838653092313255, −3.67762881048408347299678897477, −3.45747944265692077983171343238, −3.22859260101081769004929314966, −2.84643444050875121721830909544, −2.65704087481342035171870475502, −2.60913413547070431681398585092, −2.23764522966363915183976871933, −1.86529987310190334708636626818, −1.37932746312004637541757497025, −1.26464359384722805519026021915, −0.909574632935171645474804838652,
0.909574632935171645474804838652, 1.26464359384722805519026021915, 1.37932746312004637541757497025, 1.86529987310190334708636626818, 2.23764522966363915183976871933, 2.60913413547070431681398585092, 2.65704087481342035171870475502, 2.84643444050875121721830909544, 3.22859260101081769004929314966, 3.45747944265692077983171343238, 3.67762881048408347299678897477, 4.05779209180418838653092313255, 4.39176794211603451297949411961, 4.49873285644506070840628310648, 4.90738215615066665557081244035, 4.96068044842285934995907540495, 5.49871722901946486139459661329, 5.50093497655651430387291733205, 5.50792528779483325711728324327, 5.81867498413783790780442073604, 6.09424971939573400825935975273, 6.23394325079814120440714645491, 6.40967540526245773432452434970, 6.51138610755418176410632535995, 6.91706419275973185287952284097
