| L(s) = 1 | + 4-s + 25-s − 6·31-s + 2·49-s − 2·61-s − 64-s + 6·79-s + 100-s − 4·109-s − 2·121-s − 6·124-s + ⋯ |
| L(s) = 1 | + 4-s + 25-s − 6·31-s + 2·49-s − 2·61-s − 64-s + 6·79-s + 100-s − 4·109-s − 2·121-s − 6·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.162742074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162742074\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 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
−6.99475320269487844630952026441, −6.62642737977617720361907892094, −6.49160182955838938500626798135, −6.42495784055379098722209689995, −5.99184138148652764590891469838, −5.89453478103287152027248946679, −5.57197238257233679096340628042, −5.36737219805671584897445819349, −5.14105048468475931556667608079, −5.04470615060909331958343355146, −5.01144223424448215301516582066, −4.35764931623525485525467436163, −4.13804765606483582796806651783, −3.86289823568665006911219669556, −3.85641075320748535554128539130, −3.57255283615605177361288474575, −3.22998638764014325221449549178, −2.88976221028055071545930440726, −2.85903803964139520877266130723, −2.26242527132989234558910948549, −2.07937729064730881951366431317, −2.02431055089821508885389727722, −1.48230050296191406741310029786, −1.40728377320323472824327332004, −0.57880136211724557610178476678,
0.57880136211724557610178476678, 1.40728377320323472824327332004, 1.48230050296191406741310029786, 2.02431055089821508885389727722, 2.07937729064730881951366431317, 2.26242527132989234558910948549, 2.85903803964139520877266130723, 2.88976221028055071545930440726, 3.22998638764014325221449549178, 3.57255283615605177361288474575, 3.85641075320748535554128539130, 3.86289823568665006911219669556, 4.13804765606483582796806651783, 4.35764931623525485525467436163, 5.01144223424448215301516582066, 5.04470615060909331958343355146, 5.14105048468475931556667608079, 5.36737219805671584897445819349, 5.57197238257233679096340628042, 5.89453478103287152027248946679, 5.99184138148652764590891469838, 6.42495784055379098722209689995, 6.49160182955838938500626798135, 6.62642737977617720361907892094, 6.99475320269487844630952026441
