| L(s) = 1 | − 3i·9-s + (−1 + i)13-s + (−3 − 3i)17-s − 4i·29-s + (−7 − 7i)37-s − 8·41-s + 7i·49-s + (9 − 9i)53-s − 12·61-s + (11 − 11i)73-s − 9·81-s − 16i·89-s + (−13 − 13i)97-s − 2·101-s + 6i·109-s + ⋯ |
| L(s) = 1 | − i·9-s + (−0.277 + 0.277i)13-s + (−0.727 − 0.727i)17-s − 0.742i·29-s + (−1.15 − 1.15i)37-s − 1.24·41-s + i·49-s + (1.23 − 1.23i)53-s − 1.53·61-s + (1.28 − 1.28i)73-s − 81-s − 1.69i·89-s + (−1.31 − 1.31i)97-s − 0.199·101-s + 0.574i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9589279285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9589279285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + (3 + 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (7 + 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-9 + 9i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-11 + 11i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (13 + 13i)T + 97iT^{2} \) |
| 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
−9.158047673976675898491476918859, −8.505857169049932365334168389233, −7.39847823269285495455918527513, −6.77304054145758788074451628387, −5.95840057996330046255882319759, −4.95607936423897530364484505212, −4.04574733067508784466632777220, −3.10195229467936784608381047703, −1.92297272797417177597877392857, −0.35694884930754046654729483871,
1.58776063976500662580433775881, 2.62941979123255925439395993849, 3.76485928125883358608548295484, 4.81645057521060485154422430691, 5.44955610623167730756439373762, 6.55043374149371581466503007737, 7.25980775975196894870560459596, 8.226234801848473967582035547826, 8.687258232474921851433377090122, 9.767903981935925295708492327207
