| L(s) = 1 | + 0.339·3-s + i·5-s − 3.88i·7-s − 2.88·9-s − 1.54i·11-s + (3.54 + 0.660i)13-s + 0.339i·15-s − 2.86i·19-s − 1.32i·21-s − 5.42·23-s − 25-s − 2·27-s − 5.20·29-s − 6.22i·31-s − 0.524i·33-s + ⋯ |
| L(s) = 1 | + 0.196·3-s + 0.447i·5-s − 1.46i·7-s − 0.961·9-s − 0.465i·11-s + (0.983 + 0.183i)13-s + 0.0877i·15-s − 0.657i·19-s − 0.288i·21-s − 1.13·23-s − 0.200·25-s − 0.384·27-s − 0.966·29-s − 1.11i·31-s − 0.0913i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212294350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212294350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-3.54 - 0.660i)T \) |
| good | 3 | \( 1 - 0.339T + 3T^{2} \) |
| 7 | \( 1 + 3.88iT - 7T^{2} \) |
| 11 | \( 1 + 1.54iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.86iT - 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 + 6.22iT - 31T^{2} \) |
| 37 | \( 1 + 8.56iT - 37T^{2} \) |
| 41 | \( 1 + 9.08iT - 41T^{2} \) |
| 43 | \( 1 + 0.980T + 43T^{2} \) |
| 47 | \( 1 - 6.52iT - 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 4.45iT - 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 6.97iT - 67T^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + 3.43iT - 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 8.56iT - 83T^{2} \) |
| 89 | \( 1 + 17.1iT - 89T^{2} \) |
| 97 | \( 1 - 13.7iT - 97T^{2} \) |
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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
−9.699751295174514268954354070998, −8.851416210427409782911407624106, −7.959203363042301384107454194308, −7.27119078491848601440281672111, −6.30611827380466477832370035675, −5.54225561134782271052485646374, −4.03964730102890457386113108343, −3.57999037854292848947951138734, −2.22280652612071376264068446283, −0.52087717628794454071138264638,
1.69956839686137881733068151461, 2.79196807006025048337080165198, 3.85488805321423240665942762691, 5.22012886994153803640302637606, 5.75462949637089191171937094613, 6.60244552089941096674829127559, 8.148569020855830695057358092832, 8.373611855646889178885669053964, 9.213573656538685710974718380342, 9.973649897576247350732512345453
