L(s) = 1 | + (−0.991 + 0.130i)2-s + (−0.965 − 0.258i)3-s + (0.965 − 0.258i)4-s + (0.991 + 0.130i)6-s + (−0.923 + 0.382i)8-s + (0.866 + 0.499i)9-s + (−0.608 + 0.206i)11-s − 12-s + (0.866 − 0.5i)16-s + (0.130 + 0.991i)17-s + (−0.923 − 0.382i)18-s + (0.478 + 0.198i)19-s + (0.576 − 0.284i)22-s + (0.991 − 0.130i)24-s + (−0.608 + 0.793i)25-s + ⋯ |
L(s) = 1 | + (−0.991 + 0.130i)2-s + (−0.965 − 0.258i)3-s + (0.965 − 0.258i)4-s + (0.991 + 0.130i)6-s + (−0.923 + 0.382i)8-s + (0.866 + 0.499i)9-s + (−0.608 + 0.206i)11-s − 12-s + (0.866 − 0.5i)16-s + (0.130 + 0.991i)17-s + (−0.923 − 0.382i)18-s + (0.478 + 0.198i)19-s + (0.576 − 0.284i)22-s + (0.991 − 0.130i)24-s + (−0.608 + 0.793i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4509417576\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4509417576\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.991 - 0.130i)T \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 17 | \( 1 + (-0.130 - 0.991i)T \) |
good | 5 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 7 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 11 | \( 1 + (0.608 - 0.206i)T + (0.793 - 0.608i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.478 - 0.198i)T + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (0.130 + 0.991i)T^{2} \) |
| 29 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 31 | \( 1 + (0.793 + 0.608i)T^{2} \) |
| 37 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-1.88 + 0.123i)T + (0.991 - 0.130i)T^{2} \) |
| 43 | \( 1 + (0.207 + 0.158i)T + (0.258 + 0.965i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.158 - 1.20i)T + (-0.965 - 0.258i)T^{2} \) |
| 61 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 67 | \( 1 + (-1.71 - 0.991i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (-1.25 - 0.835i)T + (0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 83 | \( 1 + (-0.241 - 1.83i)T + (-0.965 + 0.258i)T^{2} \) |
| 89 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 97 | \( 1 + (1.78 + 0.117i)T + (0.991 + 0.130i)T^{2} \) |
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
−10.03077344114346344544351329392, −9.394106739291021550022309737894, −8.234710863882040696483344776866, −7.59448211571130127789693230808, −6.87294599376140987940491254906, −5.88903919701210696464437038219, −5.38485651768080113901011527534, −3.98357207890666693463558005762, −2.43415030721846935028674578172, −1.23145743650250129304671301170,
0.69581027886490647354479091577, 2.30908045068284068348947637698, 3.54929826842654646674021804065, 4.84843009184817568511887885905, 5.75653469765220149614818946087, 6.57789763486594559736276964937, 7.42772393303459292466267452461, 8.133275988050619969432885887141, 9.322233620845137964358007779971, 9.753015486394073039400639355269