| L(s) = 1 | + i·2-s − 4-s − 0.646i·5-s + (−2.44 + i)7-s − i·8-s + 0.646·10-s + (−2.79 − 1.79i)11-s + 3.09·13-s + (−1 − 2.44i)14-s + 16-s + 3.74·17-s + 5.54·19-s + 0.646i·20-s + (1.79 − 2.79i)22-s − 4·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.288i·5-s + (−0.925 + 0.377i)7-s − 0.353i·8-s + 0.204·10-s + (−0.841 − 0.540i)11-s + 0.858·13-s + (−0.267 − 0.654i)14-s + 0.250·16-s + 0.907·17-s + 1.27·19-s + 0.144i·20-s + (0.381 − 0.595i)22-s − 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.321755826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.321755826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.44 - i)T \) |
| 11 | \( 1 + (2.79 + 1.79i)T \) |
| good | 5 | \( 1 + 0.646iT - 5T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.58iT - 29T^{2} \) |
| 31 | \( 1 - 1.15iT - 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 - 5.03T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 5.03iT - 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 3.09iT - 59T^{2} \) |
| 61 | \( 1 - 9.28T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 6.32T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 - 15.9iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.593362763762765886235443742670, −8.854476416169683288136895372634, −8.144349036230312957186600809772, −7.33778460813291288732230336968, −6.38991051307602661919373967885, −5.64941013223157986264846239999, −5.03578724969221027350554088949, −3.62711239940977923416926413841, −2.94557002962312817736206494156, −1.03241790800917581745856716717,
0.69457561259195189409199766367, 2.23684250662594339649195938343, 3.28403455345688970922033534605, 3.92260474534398067938297659122, 5.19422021301334545663294515132, 5.97826120838558245981870547394, 7.06116633905250795341151243492, 7.78238794599527531614589292270, 8.726015165121194172135476407321, 9.756898723467485391864483258333
