| L(s) = 1 | + 2·5-s + 9-s + 25-s + 2·29-s − 2·41-s + 2·45-s − 49-s − 2·61-s − 4·89-s − 4·101-s + 4·109-s − 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2·5-s + 9-s + 25-s + 2·29-s − 2·41-s + 2·45-s − 49-s − 2·61-s − 4·89-s − 4·101-s + 4·109-s − 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.163827882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163827882\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−7.77253046960430871630312415362, −7.28891948476599430707801398791, −7.12899306648116238315987085785, −7.02585684708879847186149528269, −6.87893465749138443788811077912, −6.36866776584097815883184440209, −6.26115072109761079650799173810, −6.14834496880798090102101709635, −5.98189069661683671444574195067, −5.61399028497859415870112415434, −5.22837144868878712016099572722, −5.16351289662131739329509394429, −4.89485694381613977259459875115, −4.69089718754580284850893298477, −4.31183165505352995031265383958, −4.03758408865111910624035305607, −3.87276349712210614788700350242, −3.32006138099948839028592176182, −3.15618045760290732572839988046, −2.70293794684888481033897050418, −2.51949460337761851653115025744, −2.24611297220220331476510645754, −1.51478032317578336594166757865, −1.45548708902427059096773852914, −1.43148663123145049578658092340,
1.43148663123145049578658092340, 1.45548708902427059096773852914, 1.51478032317578336594166757865, 2.24611297220220331476510645754, 2.51949460337761851653115025744, 2.70293794684888481033897050418, 3.15618045760290732572839988046, 3.32006138099948839028592176182, 3.87276349712210614788700350242, 4.03758408865111910624035305607, 4.31183165505352995031265383958, 4.69089718754580284850893298477, 4.89485694381613977259459875115, 5.16351289662131739329509394429, 5.22837144868878712016099572722, 5.61399028497859415870112415434, 5.98189069661683671444574195067, 6.14834496880798090102101709635, 6.26115072109761079650799173810, 6.36866776584097815883184440209, 6.87893465749138443788811077912, 7.02585684708879847186149528269, 7.12899306648116238315987085785, 7.28891948476599430707801398791, 7.77253046960430871630312415362
