| L(s) = 1 | + 2·7-s + 9-s + 25-s + 2·31-s + 49-s + 2·63-s − 2·79-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2·7-s + 9-s + 25-s + 2·31-s + 49-s + 2·63-s − 2·79-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.843472688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.843472688\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 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
−6.32519363628123051053998326278, −6.23417187957323142510369225024, −5.78445422622444924915063157382, −5.64864481010685982334220026183, −5.55809908010840755912746469641, −5.11261790111566541857994766914, −4.94419933224220499746627567745, −4.93504306228401354458381929783, −4.66477425202364089352975642226, −4.59963254923658048700643713882, −4.30255975086990826350758572686, −4.11208642425419700829332585898, −3.83607211641063197092203292660, −3.76457067708149087298128149499, −3.37828360226792300866770150108, −3.05704747906475848402618881687, −2.83129505915355814936910632496, −2.70824659664764256926862783658, −2.30613873186312757674385410765, −2.21405427560329911039044911435, −1.77898960763403525164835240763, −1.41460542831912759723041278259, −1.39409594292326348562485309073, −1.16033213190011212783963454945, −0.66453768151723450173260740089,
0.66453768151723450173260740089, 1.16033213190011212783963454945, 1.39409594292326348562485309073, 1.41460542831912759723041278259, 1.77898960763403525164835240763, 2.21405427560329911039044911435, 2.30613873186312757674385410765, 2.70824659664764256926862783658, 2.83129505915355814936910632496, 3.05704747906475848402618881687, 3.37828360226792300866770150108, 3.76457067708149087298128149499, 3.83607211641063197092203292660, 4.11208642425419700829332585898, 4.30255975086990826350758572686, 4.59963254923658048700643713882, 4.66477425202364089352975642226, 4.93504306228401354458381929783, 4.94419933224220499746627567745, 5.11261790111566541857994766914, 5.55809908010840755912746469641, 5.64864481010685982334220026183, 5.78445422622444924915063157382, 6.23417187957323142510369225024, 6.32519363628123051053998326278
