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
−9.145210060327404657014522046302, −8.192152263829947548463624324076, −7.31021401351418615930984807572, −6.61934665433711149719088703750, −5.43752675546078448855074798468, −5.00594796657550242777241291339, −4.19175140845424130484500144937, −3.00692436585414587288626000558, −2.03873493160167086590209888837, −0.62456860391467308714401548794,
1.51795013975538364452476919141, 2.11849522457810324846160212487, 3.25072636649214047144377526473, 4.32814274497728760755599752704, 5.54291837291380504301698890474, 5.96422198649062102043075658992, 6.86159234660814441517529223325, 7.57654844002445305010518552799, 8.265151592797047247579718580752, 9.324436535970647935784959557627