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 + (−6.46 − 2.67i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (−0.578 + 1.00i)11-s + (1.73 + 2.99i)12-s + 14.8·13-s + (−9.81 + 1.29i)14-s + (−2.00 − 3.46i)16-s + (−6.30 + 10.9i)17-s + (−3.67 − 2.12i)18-s + (−16.7 + 9.65i)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 + (−0.924 − 0.381i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.0525 + 0.0910i)11-s + (0.144 + 0.249i)12-s + 1.13·13-s + (−0.700 + 0.0928i)14-s + (−0.125 − 0.216i)16-s + (−0.371 + 0.642i)17-s + (−0.204 − 0.117i)18-s + (−0.879 + 0.507i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.665i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.745 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4856711938\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4856711938\) |
\(L(2)\) |
|
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 | \( 1 + (6.46 + 2.67i)T \) |
good | 11 | \( 1 + (0.578 - 1.00i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 14.8T + 169T^{2} \) |
| 17 | \( 1 + (6.30 - 10.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (16.7 - 9.65i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-21.0 + 12.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 49.0T + 841T^{2} \) |
| 31 | \( 1 + (24.9 + 14.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (46.1 - 26.6i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 38.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 63.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-12.5 - 21.8i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-18.0 - 10.4i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (21.1 + 12.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-5.53 + 3.19i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (107. + 62.2i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 118.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (19.7 - 34.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46.4 + 80.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 5.79T + 6.88e3T^{2} \) |
| 89 | \( 1 + (131. - 75.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 144.T + 9.40e3T^{2} \) |
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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
−10.27773307431011598399222834646, −9.350547592738304167236271296839, −8.611141992604834198302236504784, −7.33667401423412291958488330156, −6.29085801208472719286536042502, −5.89921236439237487831097142351, −4.60850928676239248297421191640, −3.84096683957432677566476835656, −3.07509276990440079816047092104, −1.54034399315263090556638474804,
0.11973633570732641634645600774, 1.92129817356645248464044138327, 3.14411316240573073214175518790, 4.01887672286189656157415091488, 5.41402221886348693928755944864, 5.85105749519134572828894328461, 6.96960224770522147758072453140, 7.24002759553882339068720159293, 8.842077402596345659426090615048, 8.933302995947672080745036821781