| L(s) = 1 | + i·3-s + (2 − i)5-s + i·7-s − 9-s + 4·11-s − 2i·13-s + (1 + 2i)15-s + 2i·17-s + 2·19-s − 21-s + 6i·23-s + (3 − 4i)25-s − i·27-s − 6·29-s + 6·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.894 − 0.447i)5-s + 0.377i·7-s − 0.333·9-s + 1.20·11-s − 0.554i·13-s + (0.258 + 0.516i)15-s + 0.485i·17-s + 0.458·19-s − 0.218·21-s + 1.25i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s − 1.11·29-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61716 + 0.381761i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61716 + 0.381761i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−11.24722093977470854702683840386, −10.18195855850965947459725594849, −9.400540590551782899011016515411, −8.880243444565989468907626311507, −7.66657305421981843105497912975, −6.23716413693100665806605864196, −5.59208663617913350095037933104, −4.46122508475892966967188002404, −3.19954430820490038521196244837, −1.57802764791372544356520025922,
1.39269080588721714232351349370, 2.72924911726652814890585679813, 4.17941370642037650400736746198, 5.57745094941215196916442241126, 6.64186150562658124434502543287, 7.06270346737010173050421499299, 8.456344019713605042974294084634, 9.373846238631850193035615484065, 10.15108226649660678129842973681, 11.24019793743423503275799088355
