L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.866 − 1.5i)3-s + (0.999 − 1.73i)4-s + 2.44i·6-s + (−6.98 + 0.440i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (−3.76 + 6.52i)11-s + (−1.73 − 2.99i)12-s + 21.3·13-s + (8.24 − 5.47i)14-s + (−2.00 − 3.46i)16-s + (10.4 − 18.1i)17-s + (3.67 + 2.12i)18-s + (−20.8 + 12.0i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.288 − 0.5i)3-s + (0.249 − 0.433i)4-s + 0.408i·6-s + (−0.998 + 0.0628i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.342 + 0.593i)11-s + (−0.144 − 0.249i)12-s + 1.64·13-s + (0.588 − 0.391i)14-s + (−0.125 − 0.216i)16-s + (0.617 − 1.06i)17-s + (0.204 + 0.117i)18-s + (−1.09 + 0.634i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8584478314\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8584478314\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.98 - 0.440i)T \) |
good | 11 | \( 1 + (3.76 - 6.52i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 21.3T + 169T^{2} \) |
| 17 | \( 1 + (-10.4 + 18.1i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (20.8 - 12.0i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (4.83 - 2.79i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 9.96T + 841T^{2} \) |
| 31 | \( 1 + (-5.70 - 3.29i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-19.3 + 11.1i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 51.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (38.8 + 67.2i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-43.1 - 24.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (72.9 + 42.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (72.4 - 41.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (57.6 + 33.2i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 68.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-43.8 + 75.9i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-49.3 - 85.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 26.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + (12.0 - 6.95i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 3.69T + 9.40e3T^{2} \) |
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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
−9.307449663512447629906306407961, −8.662614489052790157303505923324, −7.84257220778207550443015152405, −6.98832257873853113711770607481, −6.28706395496238777540545223782, −5.52080787084094871137508206404, −4.00881037717733692790480688123, −2.94881419516003374534248187172, −1.72055447486433305729950196607, −0.33989603321301049270679996507,
1.21533064598380487461773830785, 2.78099236965787426305124974998, 3.51146600178088465305196720817, 4.42225727127314595122561386562, 6.07593218091869773295588125488, 6.35816199541919959663036960737, 7.85037692816427283152165999016, 8.458538750952341683555252924715, 9.140348246857090868509825770932, 9.976564293847075063116257385601