| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.343439315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.343439315\) |
| \(L(2)\) |
|
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
−8.736863452051262771660792430179, −8.393341267827296264122454520609, −7.83416983160597488404154339478, −6.91510914259903552611640721913, −6.03611147932719481261887742034, −5.30255568362274474004560266505, −4.04571638765610435984745196752, −2.54776707946392388710374118180, −1.61690828161414318234152954168, −0.77411259260226489970349166191,
1.49700553105949482919265743516, 2.87375708871137673309359839810, 4.20398320720664333977726271210, 4.66372535796963329269842578435, 5.26939804335716808910909076316, 6.47800170741559022380633770345, 7.85496651914723445621774652417, 8.412193997813358537803878121258, 9.202721245903646118742335172167, 10.08679399063259081596986801116
