| L(s) = 1 | − 2·4-s + 3·16-s − 2·49-s − 4·64-s + 4·121-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 + 191-s + 193-s + 4·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 2·49-s − 4·64-s + 4·121-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 + 191-s + 193-s + 4·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4631138334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4631138334\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.41409372856590097726043341641, −6.16345406032983810521421051557, −6.07354163042367560984761999414, −5.87576438843376658651949441175, −5.78306524062205899355110329751, −5.41834087466413072585229649494, −5.13361758867977738076735103171, −4.93841222341856648093892857258, −4.88341680713986146242717727265, −4.80689444032175955156361529305, −4.43369372567518972194928568396, −4.23040721017499197367520859141, −3.98230081103615518005565762796, −3.75818322319115444571216853515, −3.63773937410631060655005934609, −3.24714793227316224933648503366, −3.22677706503195506767150351788, −2.90396793095614900778452243259, −2.57122152479373679057078250940, −2.20431174424225376329667952216, −2.01742807936534045086476165387, −1.50929048096616552089819148766, −1.25481888847999473617822282080, −1.04773547050321697309876268420, −0.36638108559426624345244067375,
0.36638108559426624345244067375, 1.04773547050321697309876268420, 1.25481888847999473617822282080, 1.50929048096616552089819148766, 2.01742807936534045086476165387, 2.20431174424225376329667952216, 2.57122152479373679057078250940, 2.90396793095614900778452243259, 3.22677706503195506767150351788, 3.24714793227316224933648503366, 3.63773937410631060655005934609, 3.75818322319115444571216853515, 3.98230081103615518005565762796, 4.23040721017499197367520859141, 4.43369372567518972194928568396, 4.80689444032175955156361529305, 4.88341680713986146242717727265, 4.93841222341856648093892857258, 5.13361758867977738076735103171, 5.41834087466413072585229649494, 5.78306524062205899355110329751, 5.87576438843376658651949441175, 6.07354163042367560984761999414, 6.16345406032983810521421051557, 6.41409372856590097726043341641
