| L(s) = 1 | + 2.73·3-s − 0.732i·5-s + (−2 + 1.73i)7-s + 4.46·9-s + 5.46i·11-s − 4.73i·13-s − 2i·15-s + 4i·17-s + 1.26·19-s + (−5.46 + 4.73i)21-s + 5.46i·23-s + 4.46·25-s + 3.99·27-s + 6.92·29-s + 6.92·31-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 0.327i·5-s + (−0.755 + 0.654i)7-s + 1.48·9-s + 1.64i·11-s − 1.31i·13-s − 0.516i·15-s + 0.970i·17-s + 0.290·19-s + (−1.19 + 1.03i)21-s + 1.13i·23-s + 0.892·25-s + 0.769·27-s + 1.28·29-s + 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.834832553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.834832553\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 + 0.732iT - 5T^{2} \) |
| 11 | \( 1 - 5.46iT - 11T^{2} \) |
| 13 | \( 1 + 4.73iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 - 5.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 2.92iT - 41T^{2} \) |
| 43 | \( 1 + 2.53iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 - 3.80T + 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 + 2.53iT - 67T^{2} \) |
| 71 | \( 1 - 4.53iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 14.9iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−9.397014109808868046941882399303, −8.493012431694488491261084946624, −8.031052201686341892688814780371, −7.22016414844782243279662745479, −6.29759640459938582725705859473, −5.17122501258910507115592066310, −4.24827113296316988379284838738, −3.17608693914723333447999375355, −2.63469646881604824933228254168, −1.48722387404717483233693674375,
0.932347687983879312543311894919, 2.59826116250856262726436913046, 3.05697446319697971825497965314, 3.90645089385349625246229674718, 4.82385644311647470037347979052, 6.56508512973980881960217348007, 6.60710582639251470740998581017, 7.84614857077758190650529493338, 8.421976610344038630749016287078, 9.168799100118212492271122692189
