| L(s) = 1 | + 3·5-s + 7-s + 5·25-s + 3·35-s + 2·37-s − 3·41-s − 43-s − 3·47-s − 3·59-s − 2·67-s + 79-s + 3·83-s + 2·109-s + 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 5·175-s + ⋯ |
| L(s) = 1 | + 3·5-s + 7-s + 5·25-s + 3·35-s + 2·37-s − 3·41-s − 43-s − 3·47-s − 3·59-s − 2·67-s + 79-s + 3·83-s + 2·109-s + 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 5·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.629483862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.629483862\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.404131138151849460197724907370, −9.157459731001308067404370092454, −8.720675190731263099090895101008, −8.359109647841424439268837567252, −7.79656398431920366149453622911, −7.67195277408345010683696590284, −6.83540207813321222308032922865, −6.49511384356694360366106155218, −6.17117608634982345899998415721, −6.11263708493089180873039830152, −5.37956434648693243335753014504, −5.05958714357875029679961182122, −4.81335312708304757388116411562, −4.52698643413375076595754429571, −3.38674239052984311072261411925, −3.19474101201857219254464941387, −2.51342856847693899456278364096, −1.92249483388273833836499596630, −1.67705623700190681010836916939, −1.34210140320326139549277096549,
1.34210140320326139549277096549, 1.67705623700190681010836916939, 1.92249483388273833836499596630, 2.51342856847693899456278364096, 3.19474101201857219254464941387, 3.38674239052984311072261411925, 4.52698643413375076595754429571, 4.81335312708304757388116411562, 5.05958714357875029679961182122, 5.37956434648693243335753014504, 6.11263708493089180873039830152, 6.17117608634982345899998415721, 6.49511384356694360366106155218, 6.83540207813321222308032922865, 7.67195277408345010683696590284, 7.79656398431920366149453622911, 8.359109647841424439268837567252, 8.720675190731263099090895101008, 9.157459731001308067404370092454, 9.404131138151849460197724907370
