| L(s) = 1 | − 2i·3-s + 2·4-s + i·7-s − 9-s + 3·11-s − 4i·12-s − 4i·13-s + 4·16-s + 3i·17-s − 19-s + 2·21-s − 4i·27-s + 2i·28-s − 6·29-s − 4·31-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + 4-s + 0.377i·7-s − 0.333·9-s + 0.904·11-s − 1.15i·12-s − 1.10i·13-s + 16-s + 0.727i·17-s − 0.229·19-s + 0.436·21-s − 0.769i·27-s + 0.377i·28-s − 1.11·29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57003 - 0.970337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57003 - 0.970337i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−11.01599223829674273913371556322, −10.13558607609898692360040262003, −8.857929922065150000445823434414, −7.85569721733642044291143353334, −7.19338374257647227334403388816, −6.30420111822866218887948119414, −5.60467928863612811026237726981, −3.72626165952500761833576849287, −2.38918019682382824264125102261, −1.33790913205547663716745728777,
1.81850734941205597984963378329, 3.43417800691183099035068675520, 4.24437554745968537353075381868, 5.42193155285699158519675889024, 6.68555681052251664952406146848, 7.26522410276735085599114308423, 8.699800904324615898131129016567, 9.575684995004529791637221617645, 10.22769624515281842547939095267, 11.29120496476569501204698140684
