L(s) = 1 | + 1.41i·2-s + (−0.812 − 2.88i)3-s − 2.00·4-s + 2.23i·5-s + (4.08 − 1.14i)6-s − 2.64·7-s − 2.82i·8-s + (−7.68 + 4.69i)9-s − 3.16·10-s + 19.7i·11-s + (1.62 + 5.77i)12-s + 18.5·13-s − 3.74i·14-s + (6.45 − 1.81i)15-s + 4.00·16-s + 13.2i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.270 − 0.962i)3-s − 0.500·4-s + 0.447i·5-s + (0.680 − 0.191i)6-s − 0.377·7-s − 0.353i·8-s + (−0.853 + 0.521i)9-s − 0.316·10-s + 1.79i·11-s + (0.135 + 0.481i)12-s + 1.43·13-s − 0.267i·14-s + (0.430 − 0.121i)15-s + 0.250·16-s + 0.781i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.611314 + 0.806919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.611314 + 0.806919i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (0.812 + 2.88i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + 2.64T \) |
good | 11 | \( 1 - 19.7iT - 121T^{2} \) |
| 13 | \( 1 - 18.5T + 169T^{2} \) |
| 17 | \( 1 - 13.2iT - 289T^{2} \) |
| 19 | \( 1 + 5.47T + 361T^{2} \) |
| 23 | \( 1 - 34.8iT - 529T^{2} \) |
| 29 | \( 1 + 6.83iT - 841T^{2} \) |
| 31 | \( 1 + 31.4T + 961T^{2} \) |
| 37 | \( 1 + 10.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 19.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 45.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 74.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 32.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 71.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 66.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 140.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 100. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 11.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 102.T + 9.40e3T^{2} \) |
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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
−12.72256472221275513422856398034, −11.58971430675398503800926054870, −10.49295776116974259966568965219, −9.323474097193166497084866294451, −8.116186996969248990022012401722, −7.19560418842988722265036224642, −6.45834093862276747235596706012, −5.43679225298806322715836458806, −3.76669193799712002655461717824, −1.77982705880394792913036832233,
0.60716155965645057387061792344, 3.08467206455967005115956684836, 4.01992457667137295136477487889, 5.37109769405087756590356544243, 6.30530214640153260568771456856, 8.553045245073608866011085741585, 8.819160805854256507404413311814, 10.08148827552883862821941176799, 11.00361684846742200976330808286, 11.48926121941284354324595873508