| L(s) = 1 | + 2.29·3-s + (−0.507 + 4.97i)5-s − 3.40·7-s − 3.74·9-s + 3.31i·11-s − 11.3i·13-s + (−1.16 + 11.3i)15-s + 2.72i·17-s + 13.6i·19-s − 7.80·21-s − 6.07·23-s + (−24.4 − 5.04i)25-s − 29.2·27-s + 6.28·29-s + 16.4i·31-s + ⋯ |
| L(s) = 1 | + 0.763·3-s + (−0.101 + 0.994i)5-s − 0.486·7-s − 0.416·9-s + 0.301i·11-s − 0.874i·13-s + (−0.0775 + 0.759i)15-s + 0.160i·17-s + 0.720i·19-s − 0.371·21-s − 0.264·23-s + (−0.979 − 0.201i)25-s − 1.08·27-s + 0.216·29-s + 0.529i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6264685388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6264685388\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.507 - 4.97i)T \) |
| 11 | \( 1 - 3.31iT \) |
| good | 3 | \( 1 - 2.29T + 9T^{2} \) |
| 7 | \( 1 + 3.40T + 49T^{2} \) |
| 13 | \( 1 + 11.3iT - 169T^{2} \) |
| 17 | \( 1 - 2.72iT - 289T^{2} \) |
| 19 | \( 1 - 13.6iT - 361T^{2} \) |
| 23 | \( 1 + 6.07T + 529T^{2} \) |
| 29 | \( 1 - 6.28T + 841T^{2} \) |
| 31 | \( 1 - 16.4iT - 961T^{2} \) |
| 37 | \( 1 - 31.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 63.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 70.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 21.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 108. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 21.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 16.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 71.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 81.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 98.4iT - 9.40e3T^{2} \) |
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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
−10.03715150600667283425046792342, −9.809667579359847053418881279877, −8.388128843104261241567657964010, −8.049947346300431335398103966660, −6.91816640152627418882665358348, −6.20971861834974463455126805171, −5.09942800172016412047134960093, −3.52929713196886006513001806616, −3.15469164542232559669040055791, −1.96830219648644214057595954002,
0.16964168349257174503239740643, 1.78900612631058477914939461886, 3.00488515263838855856838751285, 4.00903699704768924299826260492, 5.01137934125928031823815542588, 6.02703562664626317055664972431, 7.05172255583518956085170061852, 8.114553565822419297809553169465, 8.741118064967481129394053925620, 9.320955871349909980813729515546
