L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.55 − 1.60i)5-s + (2.64 + 0.0312i)7-s − 0.999·8-s + (0.609 − 2.15i)10-s + (2.30 − 1.33i)11-s + (−1.15 + 2.00i)13-s + (1.29 + 2.30i)14-s + (−0.5 − 0.866i)16-s − 7.23i·17-s + 5.58i·19-s + (2.16 − 0.548i)20-s + (2.30 + 1.33i)22-s + (0.354 − 0.613i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.697 − 0.716i)5-s + (0.999 + 0.0118i)7-s − 0.353·8-s + (0.192 − 0.680i)10-s + (0.695 − 0.401i)11-s + (−0.321 + 0.557i)13-s + (0.346 + 0.616i)14-s + (−0.125 − 0.216i)16-s − 1.75i·17-s + 1.28i·19-s + (0.484 − 0.122i)20-s + (0.491 + 0.283i)22-s + (0.0738 − 0.127i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.932285924\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.932285924\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.55 + 1.60i)T \) |
| 7 | \( 1 + (-2.64 - 0.0312i)T \) |
good | 11 | \( 1 + (-2.30 + 1.33i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.15 - 2.00i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 7.23iT - 17T^{2} \) |
| 19 | \( 1 - 5.58iT - 19T^{2} \) |
| 23 | \( 1 + (-0.354 + 0.613i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.68 + 1.55i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.32 + 3.07i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.09iT - 37T^{2} \) |
| 41 | \( 1 + (-3.67 + 6.37i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-11.3 + 6.54i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.91 - 2.83i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 + (-1.98 + 3.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.66 + 5.00i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.96 + 2.86i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.75iT - 71T^{2} \) |
| 73 | \( 1 - 4.33T + 73T^{2} \) |
| 79 | \( 1 + (-7.29 - 12.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.25 - 1.30i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 5.66T + 89T^{2} \) |
| 97 | \( 1 + (6.43 + 11.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.017833972380184240345519074150, −8.340466900341626189658156065764, −7.51990152066932253386965273820, −7.09090710484236953968285786075, −5.78257165375117334313309493056, −5.18911913378693952753182231944, −4.28273155538081348599089631024, −3.74606751846041417832886273675, −2.20667424899166997396659568632, −0.73793991320152417517678187753,
1.19812454256052200640179816748, 2.36504483615527566901650509418, 3.36967427824293020440266679181, 4.26489142915153837737192016021, 4.88533352781806687994682614983, 6.03348932069777465879896556754, 6.86601349148762828310320722702, 7.73237734042369177893163274084, 8.432819686857210847214278406548, 9.271062368988841100607563983609