| L(s) = 1 | + 4·3-s − 8·7-s + 8·9-s + 8·17-s − 32·21-s + 12·27-s − 8·43-s + 16·49-s + 32·51-s − 8·53-s + 16·59-s − 64·63-s − 24·67-s + 16·71-s + 23·81-s + 8·103-s − 48·109-s + 40·113-s − 64·119-s − 28·121-s + 127-s − 32·129-s + 131-s + 137-s + 139-s + 64·147-s + 149-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3.02·7-s + 8/3·9-s + 1.94·17-s − 6.98·21-s + 2.30·27-s − 1.21·43-s + 16/7·49-s + 4.48·51-s − 1.09·53-s + 2.08·59-s − 8.06·63-s − 2.93·67-s + 1.89·71-s + 23/9·81-s + 0.788·103-s − 4.59·109-s + 3.76·113-s − 5.86·119-s − 2.54·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.27·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.616160006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.616160006\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 834 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 1834 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T - 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 144 T^{2} + 9314 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 12 T + 120 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_4\times C_2$ | \( 1 - 28 T^{2} - 1946 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 25522 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 8806 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_4\times C_2$ | \( 1 + 196 T^{2} + 23814 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.40618754747122412192834692838, −6.27971474564100540788488473491, −5.84488765959121776440476026372, −5.82384775796680840771419577004, −5.76765158595321343692956768986, −5.31288403578479962980625609019, −5.10313616295219844341570224798, −4.72384468186693486461135404609, −4.69927652500181606577435890075, −4.46354234011776949357906763508, −3.90929564063587839844040726507, −3.72840720237847233575000174731, −3.68453707735380182978863554314, −3.48899807429647158143863326263, −3.29090905756263309607439296516, −3.05302730140143678356196609330, −2.98127251361597979040598975363, −2.67640955063685702803443204516, −2.46786859083499354734996208455, −2.26137151504685490599491254173, −1.70365712470787722873798329228, −1.56509750721689152245785265470, −1.23317211076237883374826742726, −0.66507212839520782591072708887, −0.29065822687564782064303183452,
0.29065822687564782064303183452, 0.66507212839520782591072708887, 1.23317211076237883374826742726, 1.56509750721689152245785265470, 1.70365712470787722873798329228, 2.26137151504685490599491254173, 2.46786859083499354734996208455, 2.67640955063685702803443204516, 2.98127251361597979040598975363, 3.05302730140143678356196609330, 3.29090905756263309607439296516, 3.48899807429647158143863326263, 3.68453707735380182978863554314, 3.72840720237847233575000174731, 3.90929564063587839844040726507, 4.46354234011776949357906763508, 4.69927652500181606577435890075, 4.72384468186693486461135404609, 5.10313616295219844341570224798, 5.31288403578479962980625609019, 5.76765158595321343692956768986, 5.82384775796680840771419577004, 5.84488765959121776440476026372, 6.27971474564100540788488473491, 6.40618754747122412192834692838
