L(s) = 1 | + (0.5 − 0.866i)2-s + (1.5 + 2.59i)3-s + (−0.499 − 0.866i)4-s + (2 − 3.46i)5-s + 3·6-s + (−2.5 + 0.866i)7-s − 0.999·8-s + (−3 + 5.19i)9-s + (−1.99 − 3.46i)10-s + (0.5 + 0.866i)11-s + (1.50 − 2.59i)12-s − 13-s + (−0.500 + 2.59i)14-s + 12·15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.866 + 1.49i)3-s + (−0.249 − 0.433i)4-s + (0.894 − 1.54i)5-s + 1.22·6-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (−1 + 1.73i)9-s + (−0.632 − 1.09i)10-s + (0.150 + 0.261i)11-s + (0.433 − 0.749i)12-s − 0.277·13-s + (−0.133 + 0.694i)14-s + 3.09·15-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65038 - 0.104733i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65038 - 0.104733i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.5 + 12.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−12.96528287315446072102700425506, −12.20541195663546005208323134745, −10.55333383450835637937244143279, −9.664363201484915324423128282110, −9.286266462495648396467298288701, −8.406253260525527358347099467565, −5.89569493697794276035855828275, −4.85949936813717926247574447694, −3.90545201001924950217398459583, −2.35960317471923201559308793654,
2.39971980369470025644422909823, 3.35822139059151646624000002223, 6.01246055865774076949142190835, 6.82500268444453572766359887593, 7.17433214611602209685784023249, 8.617941701912037415993615121931, 9.735594596058672393343829737667, 11.04623671965480280605363191019, 12.51574387827071077619440490772, 13.38846371560432672660405971099