L(s) = 1 | + (−0.566 − 0.981i)5-s + (2.58 − 0.553i)7-s + (2.30 + 1.32i)11-s − 1.84i·13-s + (−0.267 + 0.463i)17-s + (2.57 − 1.48i)19-s + (−0.839 + 0.484i)23-s + (1.85 − 3.21i)25-s + 3.08i·29-s + (0.682 + 0.394i)31-s + (−2.00 − 2.22i)35-s + (−1.73 − 3.00i)37-s + 4.45·41-s + 1.69·43-s + (5.14 + 8.90i)47-s + ⋯ |
L(s) = 1 | + (−0.253 − 0.438i)5-s + (0.977 − 0.209i)7-s + (0.693 + 0.400i)11-s − 0.510i·13-s + (−0.0648 + 0.112i)17-s + (0.590 − 0.340i)19-s + (−0.175 + 0.101i)23-s + (0.371 − 0.643i)25-s + 0.572i·29-s + (0.122 + 0.0707i)31-s + (−0.339 − 0.376i)35-s + (−0.284 − 0.493i)37-s + 0.695·41-s + 0.259·43-s + (0.750 + 1.29i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.015200062\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.015200062\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.58 + 0.553i)T \) |
good | 5 | \( 1 + (0.566 + 0.981i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.30 - 1.32i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.84iT - 13T^{2} \) |
| 17 | \( 1 + (0.267 - 0.463i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.57 + 1.48i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.839 - 0.484i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.08iT - 29T^{2} \) |
| 31 | \( 1 + (-0.682 - 0.394i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.73 + 3.00i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 - 1.69T + 43T^{2} \) |
| 47 | \( 1 + (-5.14 - 8.90i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (11.9 + 6.87i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.64 - 6.31i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.74 + 1.00i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.20 + 2.08i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.3iT - 71T^{2} \) |
| 73 | \( 1 + (-7.05 - 4.07i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.58 + 4.47i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + (6.52 + 11.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.66iT - 97T^{2} \) |
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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
−8.942882874108152563455407798879, −8.082773248023418136310271556261, −7.56166562848402375134160147842, −6.67605261282542875263536997453, −5.69445002181918662381304221445, −4.80631980453170597163620162003, −4.26703094406191631368028791632, −3.16476050773618354343643654320, −1.88596876053748151561463009184, −0.846178879896908320703523445956,
1.15269456442120482371527003926, 2.25594078234033679265991670983, 3.40012309352636439267965834888, 4.23598600809274869429402374022, 5.12655934370355969564846735269, 5.99558642759525661185130415611, 6.84178327118896939403772672486, 7.60285589085110508511284384411, 8.305000878315994215567837939149, 9.075023825271540794817474041328