| L(s) = 1 | + 4-s + 2·5-s + 9-s + 2·20-s + 25-s + 36-s − 4·41-s + 2·45-s + 6·61-s − 64-s + 2·89-s + 100-s − 2·101-s + 2·109-s + 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s − 4·169-s + ⋯ |
| L(s) = 1 | + 4-s + 2·5-s + 9-s + 2·20-s + 25-s + 36-s − 4·41-s + 2·45-s + 6·61-s − 64-s + 2·89-s + 100-s − 2·101-s + 2·109-s + 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s − 4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.520554760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.520554760\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 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
−6.44799535542176559626100397305, −6.31415932888307714583375887958, −5.84905343841636601328963188542, −5.77216252705640611131519703481, −5.75200316547925334434837149483, −5.48958446628348243556367076173, −5.10707896975485862670736829097, −4.90561906296770291561543927126, −4.87992972609607544174460302970, −4.85692551329849148882312697060, −4.37217292180684241515694397461, −3.97603250503029158316758950514, −3.75575972038887152703012109411, −3.60689110370965663008295093127, −3.59232359453395060057721110132, −3.27431960975081508271738885242, −2.63356709781283595671274250554, −2.56363547412436823536181674458, −2.49506277129043202828703906605, −2.22132576484048982284946758877, −1.85236679601173407319708334497, −1.67535588114241196329373046250, −1.55745997527236133285536457076, −1.20262072308039553666549386194, −0.68791060640353429234515154877,
0.68791060640353429234515154877, 1.20262072308039553666549386194, 1.55745997527236133285536457076, 1.67535588114241196329373046250, 1.85236679601173407319708334497, 2.22132576484048982284946758877, 2.49506277129043202828703906605, 2.56363547412436823536181674458, 2.63356709781283595671274250554, 3.27431960975081508271738885242, 3.59232359453395060057721110132, 3.60689110370965663008295093127, 3.75575972038887152703012109411, 3.97603250503029158316758950514, 4.37217292180684241515694397461, 4.85692551329849148882312697060, 4.87992972609607544174460302970, 4.90561906296770291561543927126, 5.10707896975485862670736829097, 5.48958446628348243556367076173, 5.75200316547925334434837149483, 5.77216252705640611131519703481, 5.84905343841636601328963188542, 6.31415932888307714583375887958, 6.44799535542176559626100397305
