L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.41 − 1.73i)5-s + (−0.843 + 2.50i)7-s − 0.999·8-s + (−2.20 + 0.353i)10-s + (4.88 + 2.82i)11-s + (−0.902 − 1.56i)13-s + (1.74 + 1.98i)14-s + (−0.5 + 0.866i)16-s − 4.43i·17-s − 4.20i·19-s + (−0.797 + 2.08i)20-s + (4.88 − 2.82i)22-s + (−3.13 − 5.43i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.630 − 0.776i)5-s + (−0.318 + 0.947i)7-s − 0.353·8-s + (−0.698 + 0.111i)10-s + (1.47 + 0.850i)11-s + (−0.250 − 0.433i)13-s + (0.467 + 0.530i)14-s + (−0.125 + 0.216i)16-s − 1.07i·17-s − 0.964i·19-s + (−0.178 + 0.467i)20-s + (1.04 − 0.601i)22-s + (−0.653 − 1.13i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.175568796\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.175568796\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.41 + 1.73i)T \) |
| 7 | \( 1 + (0.843 - 2.50i)T \) |
good | 11 | \( 1 + (-4.88 - 2.82i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.902 + 1.56i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.43iT - 17T^{2} \) |
| 19 | \( 1 + 4.20iT - 19T^{2} \) |
| 23 | \( 1 + (3.13 + 5.43i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.728 - 0.420i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.34 + 1.35i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.95iT - 37T^{2} \) |
| 41 | \( 1 + (5.47 + 9.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (10.1 + 5.88i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.36 + 1.94i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 0.269T + 53T^{2} \) |
| 59 | \( 1 + (-2.74 - 4.74i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.75 - 3.90i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.51 + 2.02i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.50iT - 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + (-3.09 + 5.36i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.53 + 3.77i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6.72T + 89T^{2} \) |
| 97 | \( 1 + (7.17 - 12.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.908242320772544171731106977245, −8.444014180877535788909400627739, −7.15515863869608959773830834062, −6.50344770815633121987218464642, −5.32166745591047380293714877325, −4.73379822634730620729837758115, −3.90557095953169716357669813688, −2.86610227258271373660726835193, −1.80940732825740409027919322775, −0.39310169118645390131398017431,
1.46155075292880534320525172607, 3.31340788948996795726190644467, 3.74231058407515993461044331805, 4.43800935869875634430270357053, 5.87061615974619384283248509285, 6.51388523086208990599882222649, 6.99191766655968055817394232410, 8.001164929272946125424139572549, 8.421321173207366877919086883710, 9.685529765692245760827316412578