L(s) = 1 | + (0.0720 + 0.0416i)2-s + 1.73i·3-s + (−1.99 − 3.45i)4-s + 1.32i·5-s + (−0.0720 + 0.124i)6-s + (1.88 − 1.08i)7-s − 0.665i·8-s − 2.99·9-s + (−0.0549 + 0.0951i)10-s + (18.4 − 10.6i)11-s + (5.98 − 3.45i)12-s + (6.76 + 3.90i)13-s + 0.180·14-s − 2.28·15-s + (−7.95 + 13.7i)16-s + (13.7 − 23.8i)17-s + ⋯ |
L(s) = 1 | + (0.0360 + 0.0208i)2-s + 0.577i·3-s + (−0.499 − 0.864i)4-s + 0.264i·5-s + (−0.0120 + 0.0208i)6-s + (0.268 − 0.155i)7-s − 0.0831i·8-s − 0.333·9-s + (−0.00549 + 0.00951i)10-s + (1.67 − 0.967i)11-s + (0.499 − 0.288i)12-s + (0.520 + 0.300i)13-s + 0.0129·14-s − 0.152·15-s + (−0.497 + 0.861i)16-s + (0.811 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.48876 - 0.311106i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48876 - 0.311106i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73iT \) |
| 67 | \( 1 + (-27.8 - 60.9i)T \) |
good | 2 | \( 1 + (-0.0720 - 0.0416i)T + (2 + 3.46i)T^{2} \) |
| 5 | \( 1 - 1.32iT - 25T^{2} \) |
| 7 | \( 1 + (-1.88 + 1.08i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-18.4 + 10.6i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-6.76 - 3.90i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-13.7 + 23.8i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-3.95 + 6.85i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (11.7 - 20.4i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (10.4 + 18.1i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-34.8 + 20.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (26.1 - 45.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (8.87 - 5.12i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + 3.99iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (27.2 + 47.1i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 18.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 18.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + (41.8 + 24.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 71 | \( 1 + (43.8 + 75.8i)T + (-2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (22.6 - 39.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (129. - 74.7i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (72.7 - 125. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 139.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-25.9 - 14.9i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.69275136888865953990987750275, −11.31647297505676672653114320797, −10.01959535221251404148243687360, −9.358356381901825752915220614528, −8.440139800784296371557129057266, −6.77297922868150289079082267670, −5.77820779620609125654077512228, −4.61143802558028647582937677230, −3.42622132571257278737818265462, −1.07855043408494212500386945012,
1.51848726502060492819346450423, 3.47962762027463722449255886453, 4.58422368222963278460606323746, 6.15501307810876356165850089542, 7.27222956195205603237193130927, 8.370013717421350251896120710827, 8.977267838196544925201205655470, 10.29997566375776431323180094873, 11.73692248072803054807230638083, 12.40335205151601859135689683596