| L(s) = 1 | − 2·3-s − 4·5-s − 4·7-s + 2·9-s + 8·13-s + 8·15-s − 8·17-s − 8·19-s + 8·21-s + 2·25-s − 6·27-s + 16·35-s − 24·37-s − 16·39-s − 8·45-s − 8·49-s + 16·51-s + 16·57-s − 8·63-s − 32·65-s − 4·75-s + 11·81-s + 12·83-s + 32·85-s − 32·91-s + 32·95-s + 32·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s − 1.51·7-s + 2/3·9-s + 2.21·13-s + 2.06·15-s − 1.94·17-s − 1.83·19-s + 1.74·21-s + 2/5·25-s − 1.15·27-s + 2.70·35-s − 3.94·37-s − 2.56·39-s − 1.19·45-s − 8/7·49-s + 2.24·51-s + 2.11·57-s − 1.00·63-s − 3.96·65-s − 0.461·75-s + 11/9·81-s + 1.31·83-s + 3.47·85-s − 3.35·91-s + 3.28·95-s + 3.18·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07562180172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07562180172\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 2638 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 - 76 T^{2} + 3046 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 3742 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 5038 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 2998 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8574 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 8038 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 20006 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 23302 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.52211982496569215610738779668, −6.50405224968986190626471867064, −6.37568268827427035038902799427, −5.80660121952352394802535089679, −5.80213177917306486313438171353, −5.56493272807924006166457090284, −5.43821716337003280411871969304, −4.90076460901012061773843165093, −4.75621365085638684658952130187, −4.62855873528324580411610889894, −4.37142055357000575672811179246, −4.13271124916776380689088033413, −3.77822380228752773342998721276, −3.68038995722779402458139277682, −3.53417922306261054969677734272, −3.42624119283966729587300551481, −3.19993331502472225196442097619, −2.66806962355898017103246054473, −2.33119268148321605330572802028, −2.05576500429050122653053643896, −1.65209907119912943721267385622, −1.62477496664293442295210721671, −0.956765282153551266667044222471, −0.32758658200646278590944924021, −0.14537017013971243800965563307,
0.14537017013971243800965563307, 0.32758658200646278590944924021, 0.956765282153551266667044222471, 1.62477496664293442295210721671, 1.65209907119912943721267385622, 2.05576500429050122653053643896, 2.33119268148321605330572802028, 2.66806962355898017103246054473, 3.19993331502472225196442097619, 3.42624119283966729587300551481, 3.53417922306261054969677734272, 3.68038995722779402458139277682, 3.77822380228752773342998721276, 4.13271124916776380689088033413, 4.37142055357000575672811179246, 4.62855873528324580411610889894, 4.75621365085638684658952130187, 4.90076460901012061773843165093, 5.43821716337003280411871969304, 5.56493272807924006166457090284, 5.80213177917306486313438171353, 5.80660121952352394802535089679, 6.37568268827427035038902799427, 6.50405224968986190626471867064, 6.52211982496569215610738779668
