L(s) = 1 | + (0.595 + 1.03i)3-s + (−0.5 − 0.866i)5-s + 3.62·7-s + (0.790 − 1.36i)9-s + 1.46·11-s + (−0.828 + 1.43i)13-s + (0.595 − 1.03i)15-s + (0.709 + 1.22i)17-s + (−2.75 − 3.38i)19-s + (2.16 + 3.74i)21-s + (3.55 − 6.16i)23-s + (−0.499 + 0.866i)25-s + 5.45·27-s + (−0.704 + 1.22i)29-s + 2.45·31-s + ⋯ |
L(s) = 1 | + (0.343 + 0.595i)3-s + (−0.223 − 0.387i)5-s + 1.37·7-s + (0.263 − 0.456i)9-s + 0.441·11-s + (−0.229 + 0.397i)13-s + (0.153 − 0.266i)15-s + (0.171 + 0.297i)17-s + (−0.631 − 0.775i)19-s + (0.471 + 0.816i)21-s + (0.741 − 1.28i)23-s + (−0.0999 + 0.173i)25-s + 1.05·27-s + (−0.130 + 0.226i)29-s + 0.441·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.276185756\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.276185756\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (2.75 + 3.38i)T \) |
good | 3 | \( 1 + (-0.595 - 1.03i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + (0.828 - 1.43i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.709 - 1.22i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.55 + 6.16i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.704 - 1.22i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.45T + 31T^{2} \) |
| 37 | \( 1 - 0.865T + 37T^{2} \) |
| 41 | \( 1 + (1.62 + 2.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.276 - 0.478i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.67 + 2.89i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.53 - 6.12i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.67 + 4.63i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.34 - 5.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.39 + 2.40i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.50 - 7.80i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.53 + 4.39i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + (-5.84 + 10.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.44 - 9.42i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.180852283828235799535950105798, −8.816896579399066924472574705745, −8.062130632898371870781275513035, −7.10089788981487380899714160653, −6.25058604387071473736308310818, −4.85508556626251390936679590601, −4.58391552778608978742724058909, −3.63383054549466203569917912919, −2.31074033869788008170003705551, −1.04711080435630049910548461609,
1.31130260943764113955093351201, 2.15846382983342810647712414375, 3.35149446918014752896624418764, 4.50560966434648877013672142065, 5.24327179666880636021432366655, 6.34891326075618223873991445227, 7.34151789957637204149559018814, 7.82718821745606259247635471979, 8.382915843013076745582902168982, 9.405016819285104582026929385332