L(s) = 1 | − i·2-s − 4-s − 3.46·5-s + (2 + 1.73i)7-s + i·8-s + 3.46i·10-s + 6i·11-s − 1.73i·13-s + (1.73 − 2i)14-s + 16-s + 1.73·17-s + 6.92i·19-s + 3.46·20-s + 6·22-s − 3i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.54·5-s + (0.755 + 0.654i)7-s + 0.353i·8-s + 1.09i·10-s + 1.80i·11-s − 0.480i·13-s + (0.462 − 0.534i)14-s + 0.250·16-s + 0.420·17-s + 1.58i·19-s + 0.774·20-s + 1.27·22-s − 0.625i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.758534 + 0.346536i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.758534 + 0.346536i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 13 | \( 1 + 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 3iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−11.72357138072395405697011913829, −10.64622768235583505818881540233, −9.893383578432427739469225525050, −8.547563481843925220562361936009, −7.982325305409017081479046817808, −7.03983853065696803929709335219, −5.25192054419838463801047032516, −4.41590517591630275298128708970, −3.35477266239183925133216579936, −1.76967742243704503004371937678,
0.59204550983037465145953103618, 3.37595957445088214530734210099, 4.25814546414026737499904151892, 5.32647056895037338950990251867, 6.70479405118618531543716796525, 7.58097140254410139026035015797, 8.268345140362973909937409676722, 8.976154016270856584256960018024, 10.50530641072354922644130366988, 11.54217157851813920361826121261