L(s) = 1 | − 2.58i·2-s + 0.884·3-s − 4.68·4-s + i·5-s − 2.28i·6-s − 0.858i·7-s + 6.95i·8-s − 2.21·9-s + 2.58·10-s + 6.21i·11-s − 4.14·12-s − 2.22·14-s + 0.884i·15-s + 8.61·16-s − 3.95·17-s + 5.73i·18-s + ⋯ |
L(s) = 1 | − 1.82i·2-s + 0.510·3-s − 2.34·4-s + 0.447i·5-s − 0.934i·6-s − 0.324i·7-s + 2.45i·8-s − 0.739·9-s + 0.817·10-s + 1.87i·11-s − 1.19·12-s − 0.593·14-s + 0.228i·15-s + 2.15·16-s − 0.959·17-s + 1.35i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.703308 + 0.0881756i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.703308 + 0.0881756i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.58iT - 2T^{2} \) |
| 3 | \( 1 - 0.884T + 3T^{2} \) |
| 7 | \( 1 + 0.858iT - 7T^{2} \) |
| 11 | \( 1 - 6.21iT - 11T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 - 6.34iT - 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 29 | \( 1 - 0.142T + 29T^{2} \) |
| 31 | \( 1 - 5.21iT - 31T^{2} \) |
| 37 | \( 1 + 9.04iT - 37T^{2} \) |
| 41 | \( 1 - 2.38iT - 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 + 9.92iT - 47T^{2} \) |
| 53 | \( 1 - 5.45T + 53T^{2} \) |
| 59 | \( 1 - 3.55iT - 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 - 3.98iT - 67T^{2} \) |
| 71 | \( 1 - 1.45iT - 71T^{2} \) |
| 73 | \( 1 + 5.61iT - 73T^{2} \) |
| 79 | \( 1 - 0.550T + 79T^{2} \) |
| 83 | \( 1 - 4.27iT - 83T^{2} \) |
| 89 | \( 1 - 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 4.50iT - 97T^{2} \) |
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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
−10.29019923613260250597816992382, −9.703845965903171467671949666795, −8.884884128176200180803351347496, −8.011871153258402824106407152418, −6.95190447395418785779986673953, −5.46631409411145302708578513957, −4.28239227182085536181505348266, −3.66011143329098312567650120959, −2.44532033105077874416999174756, −1.82180602978540702714043220704,
0.31350793163323435624718218800, 2.84499994230179751436071640525, 4.11478786302058231803581361666, 5.17141970875430640016014915808, 5.94390431863523185244587325761, 6.55920956388664522077178373428, 7.76919048204513951426725531210, 8.472003080200036357955747913017, 8.823303485308879361128180209590, 9.507419672829403502157394014654