L(s) = 1 | − 0.746i·3-s + (−0.782 − 2.09i)5-s − i·7-s + 2.44·9-s + 5.90·11-s + 3.20i·13-s + (−1.56 + 0.584i)15-s + 2.14i·17-s + 3.56·19-s − 0.746·21-s − 3.75i·23-s + (−3.77 + 3.27i)25-s − 4.06i·27-s + 6.61·29-s − 5.79·31-s + ⋯ |
L(s) = 1 | − 0.431i·3-s + (−0.350 − 0.936i)5-s − 0.377i·7-s + 0.814·9-s + 1.78·11-s + 0.890i·13-s + (−0.403 + 0.151i)15-s + 0.521i·17-s + 0.818·19-s − 0.163·21-s − 0.782i·23-s + (−0.754 + 0.655i)25-s − 0.782i·27-s + 1.22·29-s − 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.812739937\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.812739937\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.782 + 2.09i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + 0.746iT - 3T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 - 3.20iT - 13T^{2} \) |
| 17 | \( 1 - 2.14iT - 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 3.75iT - 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + 0.623iT - 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 + 12.6iT - 43T^{2} \) |
| 47 | \( 1 + 4.31iT - 47T^{2} \) |
| 53 | \( 1 - 2.11iT - 53T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 - 1.86T + 61T^{2} \) |
| 67 | \( 1 + 6.88iT - 67T^{2} \) |
| 71 | \( 1 + 1.81T + 71T^{2} \) |
| 73 | \( 1 - 6.11iT - 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 - 13.1iT - 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.65iT - 97T^{2} \) |
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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
−9.498600628839908791686439071728, −8.884265402502901609348813418708, −8.091594716231200082642227783963, −6.95581884449227023521981574096, −6.66964111169761210823554918647, −5.32082324879117363499488600591, −4.20550803089942326726218284735, −3.81963185323592021301614892560, −1.82210962807106004193802790270, −0.987084425507636792593911859890,
1.37167028493863258099803168558, 2.99491757883076965266374970209, 3.71201293091254456039780140950, 4.68097286548359369805637613529, 5.82620119725196935175885125955, 6.76410697238644348346432994257, 7.35623639698201173983412985556, 8.353584954603952433757651096886, 9.476364358406328232044678535338, 9.813457266028578918816002033896