| L(s) = 1 | − 2·4-s + 3·16-s + 4·17-s + 4·19-s + 4·31-s − 4·47-s + 4·53-s − 4·64-s − 8·68-s − 8·76-s + 4·79-s − 4·109-s − 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 8·188-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s + 4·17-s + 4·19-s + 4·31-s − 4·47-s + 4·53-s − 4·64-s − 8·68-s − 8·76-s + 4·79-s − 4·109-s − 4·113-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 8·188-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.444429684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444429684\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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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
−6.51365430204369063828084806564, −6.34033660299625330191893121681, −6.32492904311922242424183096741, −5.90596665185178497415205119033, −5.50067882348676699322112194661, −5.48244397069680638827321513553, −5.42331498553022058175997726393, −5.24465263767617297644196765222, −4.91850566462571358267776070422, −4.85402560668873980919629450803, −4.80282883277623440770059015313, −4.26955174502167684606224047712, −3.82491265488193320850354273197, −3.81055102782744502031573539565, −3.74460014154716811243422677034, −3.40287507272241670787520641834, −3.12546505449989127700786031103, −2.94775284821295146102062262321, −2.82505653744155938864322342252, −2.57825240248484960503014651720, −1.94436799820489478805099549287, −1.23837974652002235132161079893, −1.11347230893064976083318470057, −1.08810651104447672285511050984, −0.970317249283850218835519861589,
0.970317249283850218835519861589, 1.08810651104447672285511050984, 1.11347230893064976083318470057, 1.23837974652002235132161079893, 1.94436799820489478805099549287, 2.57825240248484960503014651720, 2.82505653744155938864322342252, 2.94775284821295146102062262321, 3.12546505449989127700786031103, 3.40287507272241670787520641834, 3.74460014154716811243422677034, 3.81055102782744502031573539565, 3.82491265488193320850354273197, 4.26955174502167684606224047712, 4.80282883277623440770059015313, 4.85402560668873980919629450803, 4.91850566462571358267776070422, 5.24465263767617297644196765222, 5.42331498553022058175997726393, 5.48244397069680638827321513553, 5.50067882348676699322112194661, 5.90596665185178497415205119033, 6.32492904311922242424183096741, 6.34033660299625330191893121681, 6.51365430204369063828084806564
