L(s) = 1 | + (1.45 − 0.938i)3-s − 1.81·5-s + (−1.69 − 2.03i)7-s + (1.23 − 2.73i)9-s + 0.255i·11-s + (−5.77 + 3.33i)13-s + (−2.63 + 1.69i)15-s + (−1.99 − 3.46i)17-s + (−1.24 − 0.719i)19-s + (−4.37 − 1.37i)21-s + 5.66i·23-s − 1.71·25-s + (−0.758 − 5.14i)27-s + (−4.18 − 2.41i)29-s + (8.80 + 5.08i)31-s + ⋯ |
L(s) = 1 | + (0.840 − 0.541i)3-s − 0.810·5-s + (−0.639 − 0.768i)7-s + (0.413 − 0.910i)9-s + 0.0771i·11-s + (−1.60 + 0.925i)13-s + (−0.681 + 0.438i)15-s + (−0.484 − 0.839i)17-s + (−0.286 − 0.165i)19-s + (−0.954 − 0.299i)21-s + 1.18i·23-s − 0.343·25-s + (−0.145 − 0.989i)27-s + (−0.777 − 0.448i)29-s + (1.58 + 0.913i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5222697204\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5222697204\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.45 + 0.938i)T \) |
| 7 | \( 1 + (1.69 + 2.03i)T \) |
good | 5 | \( 1 + 1.81T + 5T^{2} \) |
| 11 | \( 1 - 0.255iT - 11T^{2} \) |
| 13 | \( 1 + (5.77 - 3.33i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.99 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.24 + 0.719i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.66iT - 23T^{2} \) |
| 29 | \( 1 + (4.18 + 2.41i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-8.80 - 5.08i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.65 - 2.86i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.10 + 8.83i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.12 - 1.94i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.97 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.97 - 2.29i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.55 + 4.42i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.60 + 4.96i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.962 + 1.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.31iT - 71T^{2} \) |
| 73 | \( 1 + (2.47 - 1.43i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.83 + 3.17i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.68 - 4.64i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.378 - 0.655i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.21 - 2.43i)T + (48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.631619143243869385902325164475, −8.658906668949671128200617550336, −7.71296513488811255922966989430, −7.10025005316179304883687270257, −6.63646234957152399604457466542, −4.93945693770360151930580978722, −4.01531628310978474134621282492, −3.15581031458247573505272336663, −1.98283899366595938942971361341, −0.19715726035180270279733269618,
2.32227461984852601233025477772, 3.06919792606697196370684784531, 4.14028446059657526276065997259, 4.96228912894417551457978719215, 6.12381579502366380196745654743, 7.23757688781920917406946048752, 8.145577724716025640354735901538, 8.543613726903365723528907353772, 9.710326781788720361202500775208, 10.06265727287559176337015087277