| L(s) = 1 | − 5-s − 1.20i·7-s + 4.86·11-s + 3.00·13-s − 5.00·17-s − 0.784i·19-s + (4.48 + 1.69i)23-s + 25-s − 2.73i·29-s + 2.49·31-s + 1.20i·35-s + 10.7i·37-s + 11.9i·41-s − 3.55i·43-s − 2.32i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.454i·7-s + 1.46·11-s + 0.832·13-s − 1.21·17-s − 0.179i·19-s + (0.935 + 0.353i)23-s + 0.200·25-s − 0.506i·29-s + 0.447·31-s + 0.203i·35-s + 1.77i·37-s + 1.87i·41-s − 0.542i·43-s − 0.339i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.079983792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079983792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-4.48 - 1.69i)T \) |
| good | 7 | \( 1 + 1.20iT - 7T^{2} \) |
| 11 | \( 1 - 4.86T + 11T^{2} \) |
| 13 | \( 1 - 3.00T + 13T^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 19 | \( 1 + 0.784iT - 19T^{2} \) |
| 29 | \( 1 + 2.73iT - 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 - 10.7iT - 37T^{2} \) |
| 41 | \( 1 - 11.9iT - 41T^{2} \) |
| 43 | \( 1 + 3.55iT - 43T^{2} \) |
| 47 | \( 1 + 2.32iT - 47T^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 - 3.40iT - 59T^{2} \) |
| 61 | \( 1 - 1.35iT - 61T^{2} \) |
| 67 | \( 1 - 5.73iT - 67T^{2} \) |
| 71 | \( 1 - 2.08iT - 71T^{2} \) |
| 73 | \( 1 - 6.31T + 73T^{2} \) |
| 79 | \( 1 + 10.9iT - 79T^{2} \) |
| 83 | \( 1 - 9.22T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
| 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
−7.894270938921201181119007987853, −6.91339982757369541725773645006, −6.64200992749556585702841899081, −5.94841736280727789912913586631, −4.79025592658954518330737650991, −4.30998068885675887670597280747, −3.61300013909122763297837059613, −2.84477076839284333163329456921, −1.59064197512165373716754524623, −0.849951519019956819311425660692,
0.65835747464536920750118870179, 1.69932119826033060047424080337, 2.61005471290717133906452632100, 3.69266804980336909256198144167, 4.04799847546434753678700505445, 4.95055919401935551748314081635, 5.76855633871753283103533534206, 6.57831495038304912703118976933, 6.87676391802091633022169707642, 7.81446598850530279287737597045
