| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s − 1.41·17-s + 1.41·23-s + 25-s − 27-s + 36-s + 1.41·47-s − 48-s + 49-s + 1.41·51-s − 1.41·59-s + 64-s − 1.41·68-s − 1.41·69-s − 75-s + 81-s − 1.41·83-s + 1.41·89-s + 1.41·92-s + 100-s + 1.41·101-s − 108-s − 1.41·113-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 16-s − 1.41·17-s + 1.41·23-s + 25-s − 27-s + 36-s + 1.41·47-s − 48-s + 49-s + 1.41·51-s − 1.41·59-s + 64-s − 1.41·68-s − 1.41·69-s − 75-s + 81-s − 1.41·83-s + 1.41·89-s + 1.41·92-s + 100-s + 1.41·101-s − 108-s − 1.41·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.241472223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.241472223\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 1321 | \( 1 + T \) |
| good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.707023342066909903161638849332, −7.57271153099236985996189801022, −7.01592952548855879925504295269, −6.51103044291409110986515632503, −5.78048495180866529098807243673, −4.98023842266590166498287244126, −4.22234867707429698719720276281, −3.06709772840226929070389955271, −2.13488831069140154661784255432, −1.02782717908849550474185863782,
1.02782717908849550474185863782, 2.13488831069140154661784255432, 3.06709772840226929070389955271, 4.22234867707429698719720276281, 4.98023842266590166498287244126, 5.78048495180866529098807243673, 6.51103044291409110986515632503, 7.01592952548855879925504295269, 7.57271153099236985996189801022, 8.707023342066909903161638849332
