L(s) = 1 | + (−0.692 − 2.12i)5-s + (−0.357 − 0.357i)7-s − 5.38i·11-s + (3.74 − 3.74i)13-s + (3.16 − 3.16i)17-s − 1.26i·19-s + (0.707 + 0.707i)23-s + (−4.04 + 2.94i)25-s + 7.94·29-s + 10.7·31-s + (−0.512 + 1.00i)35-s + (7.93 + 7.93i)37-s − 0.157i·41-s + (−6.80 + 6.80i)43-s + (3.07 − 3.07i)47-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.950i)5-s + (−0.135 − 0.135i)7-s − 1.62i·11-s + (1.04 − 1.04i)13-s + (0.768 − 0.768i)17-s − 0.290i·19-s + (0.147 + 0.147i)23-s + (−0.808 + 0.588i)25-s + 1.47·29-s + 1.93·31-s + (−0.0867 + 0.170i)35-s + (1.30 + 1.30i)37-s − 0.0245i·41-s + (−1.03 + 1.03i)43-s + (0.448 − 0.448i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.901799418\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.901799418\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.692 + 2.12i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (0.357 + 0.357i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.38iT - 11T^{2} \) |
| 13 | \( 1 + (-3.74 + 3.74i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.16 + 3.16i)T - 17iT^{2} \) |
| 19 | \( 1 + 1.26iT - 19T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + (-7.93 - 7.93i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.157iT - 41T^{2} \) |
| 43 | \( 1 + (6.80 - 6.80i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.07 + 3.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.49 + 5.49i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.30T + 59T^{2} \) |
| 61 | \( 1 + 9.56T + 61T^{2} \) |
| 67 | \( 1 + (7.26 + 7.26i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.286iT - 71T^{2} \) |
| 73 | \( 1 + (-0.811 + 0.811i)T - 73iT^{2} \) |
| 79 | \( 1 - 9.01iT - 79T^{2} \) |
| 83 | \( 1 + (3.24 + 3.24i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + (-5.54 - 5.54i)T + 97iT^{2} \) |
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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
−8.210841484877613094989783591297, −7.81498159142303640587063054626, −6.45569789227411949324993712356, −6.03936839171274693669799158916, −5.11772109405440288212720565978, −4.52733467699001766631045522445, −3.28102069107421193039278012258, −3.03131036391427813190813449157, −1.13265642951609928928058662208, −0.67304024631030393689374525587,
1.34837165300377799624008534323, 2.34591034154854839359495040269, 3.22190219024981997162739892302, 4.21399201393285280724814830843, 4.62653651173156338510146868785, 6.18484331907388486443667072096, 6.25373093208977224970639659887, 7.26302128359577810090699559739, 7.78175693509615353229482249218, 8.619118796929430908608194762850