| L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s − 3·11-s + 4i·13-s − 14-s + 16-s − 2·19-s + 3i·22-s + 6i·23-s + 4·26-s + i·28-s − 6·29-s + 5·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s − 0.904·11-s + 1.10i·13-s − 0.267·14-s + 0.250·16-s − 0.458·19-s + 0.639i·22-s + 1.25i·23-s + 0.784·26-s + 0.188i·28-s − 1.11·29-s + 0.898·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7982387923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7982387923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 3iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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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
−9.823414658854538280547746605256, −9.137288100363627401041529411461, −8.194933840539761175864112400247, −7.44021300600891822192052227329, −6.46292514303513538605773681400, −5.39858804121206894893915243031, −4.51526587989759723099959846755, −3.64835465306340412037549005718, −2.54491434853310812021989086510, −1.43727115772932601495068081392,
0.32856178543528485419301007217, 2.26924076811293193760293267965, 3.36421049332788707906543866948, 4.58218442781802802213296824214, 5.40739520066832524713666981185, 6.07149121994267881153810221377, 7.06599363266214128708217466366, 7.892586595164355605479553391953, 8.492687377859773767429612702981, 9.267322431554275110838769388705
