L(s) = 1 | − i·3-s + (2 + i)5-s + i·7-s + 2·9-s − 3·11-s + i·13-s + (1 − 2i)15-s + 7i·17-s + 21-s + 6i·23-s + (3 + 4i)25-s − 5i·27-s − 5·29-s − 2·31-s + 3i·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.894 + 0.447i)5-s + 0.377i·7-s + 0.666·9-s − 0.904·11-s + 0.277i·13-s + (0.258 − 0.516i)15-s + 1.69i·17-s + 0.218·21-s + 1.25i·23-s + (0.600 + 0.800i)25-s − 0.962i·27-s − 0.928·29-s − 0.359·31-s + 0.522i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.855816271\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.855816271\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−9.324436535970647935784959557627, −8.265151592797047247579718580752, −7.57654844002445305010518552799, −6.86159234660814441517529223325, −5.96422198649062102043075658992, −5.54291837291380504301698890474, −4.32814274497728760755599752704, −3.25072636649214047144377526473, −2.11849522457810324846160212487, −1.51795013975538364452476919141,
0.62456860391467308714401548794, 2.03873493160167086590209888837, 3.00692436585414587288626000558, 4.19175140845424130484500144937, 5.00594796657550242777241291339, 5.43752675546078448855074798468, 6.61934665433711149719088703750, 7.31021401351418615930984807572, 8.192152263829947548463624324076, 9.145210060327404657014522046302