L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.23i·5-s + 7.96i·7-s − 2.82·8-s + 3.16i·10-s − 13.6i·11-s − 14.0·13-s − 11.2i·14-s + 4.00·16-s + 17.2i·17-s − 26.4i·19-s − 4.47i·20-s + 19.3i·22-s + (11.7 − 19.7i)23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.447i·5-s + 1.13i·7-s − 0.353·8-s + 0.316i·10-s − 1.24i·11-s − 1.07·13-s − 0.804i·14-s + 0.250·16-s + 1.01i·17-s − 1.39i·19-s − 0.223i·20-s + 0.878i·22-s + (0.511 − 0.859i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2214333188\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2214333188\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-11.7 + 19.7i)T \) |
good | 7 | \( 1 - 7.96iT - 49T^{2} \) |
| 11 | \( 1 + 13.6iT - 121T^{2} \) |
| 13 | \( 1 + 14.0T + 169T^{2} \) |
| 17 | \( 1 - 17.2iT - 289T^{2} \) |
| 19 | \( 1 + 26.4iT - 361T^{2} \) |
| 29 | \( 1 - 54.4T + 841T^{2} \) |
| 31 | \( 1 + 22.1T + 961T^{2} \) |
| 37 | \( 1 - 32.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 9.59T + 1.68e3T^{2} \) |
| 43 | \( 1 - 65.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 82.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 21.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 105.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 33.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 34.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 65.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 126.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 45.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 10.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 20.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 146. iT - 9.40e3T^{2} \) |
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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
−9.069099232588786951757658035773, −8.511790787639017640780239653030, −8.115335565700810555526056209358, −6.85334878607605961286704868000, −6.25177319718870118098205126059, −5.32154349981890987850775830689, −4.59173239066301292596087021293, −3.06606474325171194607660549480, −2.47020176497169018887895722818, −1.13359750021912749726273429766,
0.07606970809206546137086778271, 1.42021474121308488537589825840, 2.45991247212555229960227639961, 3.54004562284631853584368437148, 4.52613740462862065301678947257, 5.41121208642530240418529633499, 6.66378392663118217961106229153, 7.34255402125265526076550534566, 7.49076651909367667554740877241, 8.620863725691949507772019675903