L(s) = 1 | − 2.95i·3-s − 3.95i·7-s − 5.74·9-s − 0.957·11-s − 2.74i·13-s − 5.74i·17-s + 6.74·19-s − 11.7·21-s + i·23-s + 8.12i·27-s − 5.21·29-s − 5.95·31-s + 2.83i·33-s − 9.12i·37-s − 8.12·39-s + ⋯ |
L(s) = 1 | − 1.70i·3-s − 1.49i·7-s − 1.91·9-s − 0.288·11-s − 0.761i·13-s − 1.39i·17-s + 1.54·19-s − 2.55·21-s + 0.208i·23-s + 1.56i·27-s − 0.967·29-s − 1.07·31-s + 0.493i·33-s − 1.50i·37-s − 1.30·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.375339526\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.375339526\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 2.95iT - 3T^{2} \) |
| 7 | \( 1 + 3.95iT - 7T^{2} \) |
| 11 | \( 1 + 0.957T + 11T^{2} \) |
| 13 | \( 1 + 2.74iT - 13T^{2} \) |
| 17 | \( 1 + 5.74iT - 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 + 9.12iT - 37T^{2} \) |
| 41 | \( 1 - 0.252T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 5.49iT - 47T^{2} \) |
| 53 | \( 1 + 7.12iT - 53T^{2} \) |
| 59 | \( 1 - 4.78T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 9.12iT - 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 - 12.3iT - 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 0.704iT - 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−7.60749891899960917536263720206, −7.21380932803335607428762624226, −6.76997925782222700329642245181, −5.60330963634968145901546893020, −5.19355378991471582778809749934, −3.82246733063857476341964657819, −3.07569923639357891903380944034, −2.08696144199675852094644105325, −1.00221515237025462256919756739, −0.43215039200325439924888444786,
1.81166787437212012243185777794, 2.83631217822550075622880749205, 3.58146424281946542996058199995, 4.30356250060874856613590651342, 5.27805928863159772762486076728, 5.53390656996627086922856834110, 6.33377654175968675279954334907, 7.50487884450880196741006073596, 8.461226086453458510116896568626, 8.932785431146564220040795085507