| L(s) = 1 | + 8·13-s + 8·25-s + 4·37-s + 28·49-s − 20·61-s − 32·73-s − 32·97-s − 40·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.21·13-s + 8/5·25-s + 0.657·37-s + 4·49-s − 2.56·61-s − 3.74·73-s − 3.24·97-s − 3.83·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.010751720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010751720\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 16 T^{2} - 33 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^3$ | \( 1 + 40 T^{2} + 759 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 + 160 T^{2} + 17679 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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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
−5.61340406930376714279081367512, −5.60344334913881434955640151509, −5.56639137973970488473870387304, −5.36851200526506299167173925973, −5.02955203444385962925355234514, −4.82710219100632948855374629565, −4.41540346631628875617819421401, −4.39427264034013398838529129878, −4.25891699897628032849144607366, −3.99001146740615191046411617626, −3.94266072376654101774795920924, −3.68273997889458812582183680557, −3.37298897056664900958372637250, −3.07988704666058395521660110139, −2.92563254349373294463243580490, −2.85241532655712777592586736071, −2.54267531726247469834098707371, −2.33851822380639109637067239915, −2.10303912291642183599488837109, −1.48347402854187972234114226856, −1.41017383725277621549410799926, −1.26429381852745365632434878234, −1.12998799145291547729561893251, −0.68500827244781171931718963901, −0.12334665747166157127810150687,
0.12334665747166157127810150687, 0.68500827244781171931718963901, 1.12998799145291547729561893251, 1.26429381852745365632434878234, 1.41017383725277621549410799926, 1.48347402854187972234114226856, 2.10303912291642183599488837109, 2.33851822380639109637067239915, 2.54267531726247469834098707371, 2.85241532655712777592586736071, 2.92563254349373294463243580490, 3.07988704666058395521660110139, 3.37298897056664900958372637250, 3.68273997889458812582183680557, 3.94266072376654101774795920924, 3.99001146740615191046411617626, 4.25891699897628032849144607366, 4.39427264034013398838529129878, 4.41540346631628875617819421401, 4.82710219100632948855374629565, 5.02955203444385962925355234514, 5.36851200526506299167173925973, 5.56639137973970488473870387304, 5.60344334913881434955640151509, 5.61340406930376714279081367512
