| L(s) = 1 | + 4.94i·3-s − 13.1·7-s − 15.4·9-s + (−3.18 − 10.5i)11-s − 14.6·13-s + 9.90·17-s + 11.5i·19-s − 65.1i·21-s + 14.6i·23-s − 32.1i·27-s + 8.28i·29-s + 53.2·31-s + (52.1 − 15.7i)33-s + 41.1i·37-s − 72.4i·39-s + ⋯ |
| L(s) = 1 | + 1.64i·3-s − 1.87·7-s − 1.72·9-s + (−0.289 − 0.957i)11-s − 1.12·13-s + 0.582·17-s + 0.607i·19-s − 3.10i·21-s + 0.634i·23-s − 1.18i·27-s + 0.285i·29-s + 1.71·31-s + (1.57 − 0.477i)33-s + 1.11i·37-s − 1.85i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6405652607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6405652607\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (3.18 + 10.5i)T \) |
| good | 3 | \( 1 - 4.94iT - 9T^{2} \) |
| 7 | \( 1 + 13.1T + 49T^{2} \) |
| 13 | \( 1 + 14.6T + 169T^{2} \) |
| 17 | \( 1 - 9.90T + 289T^{2} \) |
| 19 | \( 1 - 11.5iT - 361T^{2} \) |
| 23 | \( 1 - 14.6iT - 529T^{2} \) |
| 29 | \( 1 - 8.28iT - 841T^{2} \) |
| 31 | \( 1 - 53.2T + 961T^{2} \) |
| 37 | \( 1 - 41.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 74.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 30.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 0.697iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 17.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 6.29T + 3.48e3T^{2} \) |
| 61 | \( 1 + 50.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 81.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 62.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 11.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 79.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 81.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 136.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 80.2iT - 9.40e3T^{2} \) |
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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
−9.941706261431031831422797661398, −9.135126164177396004236739083154, −8.259042876507364124346623160668, −7.03559713345586852718110569472, −5.98701198345747393995829059721, −5.37486380682190939803452119422, −4.25597609932100687506141567046, −3.33335931530388393114760228757, −2.86756273832959321884979242620, −0.25910452302048472504793227672,
0.833625995340829905585835841461, 2.43396652142583862187761731961, 2.88728308548349038209161104988, 4.48031082524121305754280478148, 5.81758541285532750769399351834, 6.51102479481860856475378404998, 7.16278388493448241589218530185, 7.67865029123086457568248207591, 8.805302909836885145197982024679, 9.772246118117652285484617292589
