L(s) = 1 | + (1.62 − 0.935i)5-s + 2.07i·7-s − 5.79·11-s + (3.25 − 5.63i)13-s + (3.56 − 2.06i)17-s + (−2.66 − 3.44i)19-s + (−2.02 + 3.50i)23-s + (−0.748 + 1.29i)25-s + (−2.90 − 1.67i)29-s − 5.15i·31-s + (1.94 + 3.36i)35-s − 6.99·37-s + (−9.12 + 5.27i)41-s + (0.638 − 0.368i)43-s + (4.95 − 8.58i)47-s + ⋯ |
L(s) = 1 | + (0.724 − 0.418i)5-s + 0.785i·7-s − 1.74·11-s + (0.901 − 1.56i)13-s + (0.865 − 0.499i)17-s + (−0.611 − 0.791i)19-s + (−0.422 + 0.731i)23-s + (−0.149 + 0.259i)25-s + (−0.538 − 0.310i)29-s − 0.925i·31-s + (0.328 + 0.569i)35-s − 1.15·37-s + (−1.42 + 0.823i)41-s + (0.0973 − 0.0561i)43-s + (0.723 − 1.25i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.146298952\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.146298952\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2.66 + 3.44i)T \) |
good | 5 | \( 1 + (-1.62 + 0.935i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 2.07iT - 7T^{2} \) |
| 11 | \( 1 + 5.79T + 11T^{2} \) |
| 13 | \( 1 + (-3.25 + 5.63i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.56 + 2.06i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.02 - 3.50i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.90 + 1.67i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.15iT - 31T^{2} \) |
| 37 | \( 1 + 6.99T + 37T^{2} \) |
| 41 | \( 1 + (9.12 - 5.27i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.638 + 0.368i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.95 + 8.58i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.83 + 2.79i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.37 + 2.38i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.20 + 12.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.3 + 5.99i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.81 - 11.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.73 - 8.20i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.40 + 5.43i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.44T + 83T^{2} \) |
| 89 | \( 1 + (1.89 + 1.09i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.380 + 0.658i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.325809662337999486424017867263, −8.132109633124210116947932223214, −7.13359594811274406244498631675, −5.92180109881337712220692749820, −5.43098127951393088354739460016, −5.10426843320426967628275561302, −3.55174138924820097508935116424, −2.74978376203929447082761193912, −1.84830176240955420784167993570, −0.34465394670868594344294591993,
1.49195866569634527181607505307, 2.34026081071871599958523958442, 3.49466112670177888488160788065, 4.26876202928572525612040846909, 5.30154968745246606263194881655, 6.06378825507112854033921943280, 6.74019365067032461166866913920, 7.55007721220869105198977716940, 8.307236556328889499591053306938, 9.020186834385248028386409847790