| 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
−9.973649897576247350732512345453, −9.213573656538685710974718380342, −8.373611855646889178885669053964, −8.148569020855830695057358092832, −6.60244552089941096674829127559, −5.75462949637089191171937094613, −5.22012886994153803640302637606, −3.85488805321423240665942762691, −2.79196807006025048337080165198, −1.69956839686137881733068151461,
0.52087717628794454071138264638, 2.22280652612071376264068446283, 3.57999037854292848947951138734, 4.03964730102890457386113108343, 5.54225561134782271052485646374, 6.30611827380466477832370035675, 7.27119078491848601440281672111, 7.959203363042301384107454194308, 8.851416210427409782911407624106, 9.699751295174514268954354070998
