| L(s) = 1 | − 2i·4-s + (−3.67 + 3.67i)7-s + 3i·9-s − 4.35·11-s − 4·16-s + (1.32 − 1.32i)17-s + 4.35i·19-s + (2.35 + 2.35i)23-s + (7.35 + 7.35i)28-s + 6·36-s + (−6.03 − 6.03i)43-s + 8.71i·44-s + (−8.67 + 8.67i)47-s − 20.0i·49-s − 4.35·61-s + ⋯ |
| L(s) = 1 | − i·4-s + (−1.39 + 1.39i)7-s + i·9-s − 1.31·11-s − 16-s + (0.320 − 0.320i)17-s + 0.999i·19-s + (0.491 + 0.491i)23-s + (1.39 + 1.39i)28-s + 36-s + (−0.920 − 0.920i)43-s + 1.31i·44-s + (−1.26 + 1.26i)47-s − 2.86i·49-s − 0.558·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.260533 + 0.467294i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.260533 + 0.467294i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 2 | \( 1 + 2iT^{2} \) |
| 3 | \( 1 - 3iT^{2} \) |
| 7 | \( 1 + (3.67 - 3.67i)T - 7iT^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (-1.32 + 1.32i)T - 17iT^{2} \) |
| 23 | \( 1 + (-2.35 - 2.35i)T + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (6.03 + 6.03i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.67 - 8.67i)T - 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4.35T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (1.03 + 1.03i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-12.3 - 12.3i)T + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97iT^{2} \) |
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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
−11.16617735249840585536003821803, −10.22344943577430790637512610936, −9.731342893085473341629824212403, −8.764032885773144063008072124167, −7.70708095785568333377653997360, −6.44941810587166817618596429198, −5.58508075548018664640411316183, −5.05351519780631410895771827121, −3.10852318202880776455356011934, −2.09818360877445881677812392802,
0.30374544067046084425580504143, 2.94521458717871129049045391629, 3.56657949847332998137500340845, 4.70778975654487276188941393264, 6.37954543982250408774241704807, 7.01987432224481562987139020813, 7.84509330931984383084080124554, 8.941193147482263649774053666819, 9.865790766024906865822008375870, 10.60086460262943919192949000977
