L(s) = 1 | + (−0.707 − 0.707i)2-s + (−1.41 − i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (0.292 + 1.70i)6-s + 2·7-s + (0.707 − 0.707i)8-s + (1.00 + 2.82i)9-s + 1.00·10-s + (1.00 − 1.41i)12-s + (1 + i)13-s + (−1.41 − 1.41i)14-s + (1.70 − 0.292i)15-s − 1.00·16-s + (−4.24 + 4.24i)17-s + (1.29 − 2.70i)18-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.816 − 0.577i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (0.119 + 0.696i)6-s + 0.755·7-s + (0.250 − 0.250i)8-s + (0.333 + 0.942i)9-s + 0.316·10-s + (0.288 − 0.408i)12-s + (0.277 + 0.277i)13-s + (−0.377 − 0.377i)14-s + (0.440 − 0.0756i)15-s − 0.250·16-s + (−1.02 + 1.02i)17-s + (0.304 − 0.638i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2631128717\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2631128717\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (1 + 6i)T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.24 - 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 + (3 + 3i)T + 19iT^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + (4.24 + 4.24i)T + 29iT^{2} \) |
| 31 | \( 1 + (7 - 7i)T - 31iT^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + (-7 - 7i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (5.65 - 5.65i)T - 59iT^{2} \) |
| 61 | \( 1 + (9 - 9i)T - 61iT^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + (-3 - 3i)T + 79iT^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + (-1.41 - 1.41i)T + 89iT^{2} \) |
| 97 | \( 1 + (-7 - 7i)T + 97iT^{2} \) |
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
−10.57250426468755360135818824366, −9.138221323120998059259512087705, −8.476743266371307245838190761855, −7.58544964106855497942900675124, −6.86657709383484072232327963276, −6.02214866294072957680886457212, −4.81085766092619711605133367371, −4.00067505904214859449536790195, −2.42410431095304998578483844774, −1.47071880303995430767020659839,
0.15665999798825613374994193693, 1.69605837434562825066105169149, 3.60068685911881274103624927916, 4.68132313320446542678656033379, 5.26837913012982876458972703885, 6.19995369563461895392941089814, 7.14135111731970326643039348502, 7.932072474015699313695133216875, 8.947831170893785557855074771772, 9.411344684351459884532897647738