L(s) = 1 | + (1.38 − 1.43i)2-s + (−0.142 − 3.99i)4-s − 1.77·5-s − 3.79i·7-s + (−5.94 − 5.34i)8-s + (−2.46 + 2.55i)10-s − 9.58i·11-s + 22.4·13-s + (−5.45 − 5.26i)14-s + (−15.9 + 1.13i)16-s − 11.9·17-s + 4.35i·19-s + (0.252 + 7.09i)20-s + (−13.7 − 13.3i)22-s − 30.8i·23-s + ⋯ |
L(s) = 1 | + (0.694 − 0.719i)2-s + (−0.0355 − 0.999i)4-s − 0.355·5-s − 0.541i·7-s + (−0.743 − 0.668i)8-s + (−0.246 + 0.255i)10-s − 0.871i·11-s + 1.72·13-s + (−0.389 − 0.376i)14-s + (−0.997 + 0.0709i)16-s − 0.705·17-s + 0.229i·19-s + (0.0126 + 0.354i)20-s + (−0.626 − 0.604i)22-s − 1.34i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0355i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.794046113\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.794046113\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 + 1.43i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 5 | \( 1 + 1.77T + 25T^{2} \) |
| 7 | \( 1 + 3.79iT - 49T^{2} \) |
| 11 | \( 1 + 9.58iT - 121T^{2} \) |
| 13 | \( 1 - 22.4T + 169T^{2} \) |
| 17 | \( 1 + 11.9T + 289T^{2} \) |
| 23 | \( 1 + 30.8iT - 529T^{2} \) |
| 29 | \( 1 + 36.2T + 841T^{2} \) |
| 31 | \( 1 - 40.0iT - 961T^{2} \) |
| 37 | \( 1 + 58.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 34.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 41.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 22.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 62.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 79.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 127. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 107.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 99.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 18.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 50.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + 74.6T + 9.40e3T^{2} \) |
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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
−10.27136166144445740618460951609, −8.935807465846547846735922214710, −8.388682374956643833036026105229, −6.92150476944376581498242426098, −6.15239325754943551458750085876, −5.16276958564050107480490868743, −3.88607166741145991339215485044, −3.47649279334430270434337584341, −1.85785904634471732407524881451, −0.49542758251978325368722225432,
1.99703313677000658388040269249, 3.49584327508304106662875895971, 4.21752040308982983958103019317, 5.43126535612959139520793491291, 6.13766761526887142774261130689, 7.15007780501681619471144446084, 7.940575383715870018657075995905, 8.819345592466460585180445039536, 9.565618379932248952205009919860, 11.07797322825168056924149946499