| L(s) = 1 | + (0.587 − 0.809i)2-s + (0.166 − 0.0542i)3-s + (−0.309 − 0.951i)4-s + (−1.44 + 1.70i)5-s + (0.0542 − 0.166i)6-s − 1.27i·7-s + (−0.951 − 0.309i)8-s + (−2.40 + 1.74i)9-s + (0.530 + 2.17i)10-s + (3.52 + 2.56i)11-s + (−0.103 − 0.142i)12-s + (−2.51 − 3.46i)13-s + (−1.03 − 0.748i)14-s + (−0.148 + 0.363i)15-s + (−0.809 + 0.587i)16-s + (−0.0930 − 0.0302i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (0.0964 − 0.0313i)3-s + (−0.154 − 0.475i)4-s + (−0.646 + 0.762i)5-s + (0.0221 − 0.0681i)6-s − 0.481i·7-s + (−0.336 − 0.109i)8-s + (−0.800 + 0.581i)9-s + (0.167 + 0.686i)10-s + (1.06 + 0.773i)11-s + (−0.0297 − 0.0410i)12-s + (−0.698 − 0.961i)13-s + (−0.275 − 0.200i)14-s + (−0.0384 + 0.0937i)15-s + (−0.202 + 0.146i)16-s + (−0.0225 − 0.00733i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.881235 - 0.267446i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.881235 - 0.267446i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 + (1.44 - 1.70i)T \) |
| good | 3 | \( 1 + (-0.166 + 0.0542i)T + (2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 1.27iT - 7T^{2} \) |
| 11 | \( 1 + (-3.52 - 2.56i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (2.51 + 3.46i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.0930 + 0.0302i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.103 + 0.317i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.71 + 5.10i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.44 - 4.44i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.48 + 7.64i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.41 - 3.32i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (9.24 - 6.71i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 + (11.1 - 3.61i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.841 - 0.273i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.15 + 4.47i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.35 - 0.985i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.11 - 0.361i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (0.728 + 2.24i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.945 - 1.30i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.51 + 7.73i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-7.05 - 2.29i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.38 - 1.00i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.28 + 2.04i)T + (78.4 - 57.0i)T^{2} \) |
| 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
−14.91740012146286716857703545045, −14.56812392481623863456295864210, −13.16484213145518168446865475492, −11.90536134636235312731550016194, −10.97639297525911634176112648404, −9.861801868620983333904419695450, −8.067541962834494240159915205807, −6.64754361469975101462280608402, −4.64796696477500964077998786232, −2.95377204869347223887838558003,
3.69244566039612321076650642740, 5.33270925583842778002158332295, 6.83341218816448078998063746876, 8.516577919528192480902075830540, 9.201010906699680665544296975908, 11.66138899626714894133375470798, 12.05993398509226092133453374974, 13.63986445175225068202422333592, 14.64030610090816322598619222652, 15.62711678417349749549761130759
