L(s) = 1 | + (−1 + 1.41i)3-s + (0.5 + 0.866i)5-s + (−0.724 + 1.25i)7-s + (−1.00 − 2.82i)9-s + (−1.72 + 2.98i)11-s + (1.94 + 3.37i)13-s + (−1.72 − 0.158i)15-s − 4.89·17-s − 4·19-s + (−1.05 − 2.28i)21-s + (−0.275 − 0.476i)23-s + (2 − 3.46i)25-s + (5.00 + 1.41i)27-s + (−4.94 + 8.57i)29-s + (3.72 + 6.45i)31-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (0.223 + 0.387i)5-s + (−0.273 + 0.474i)7-s + (−0.333 − 0.942i)9-s + (−0.520 + 0.900i)11-s + (0.540 + 0.936i)13-s + (−0.445 − 0.0410i)15-s − 1.18·17-s − 0.917·19-s + (−0.229 − 0.497i)21-s + (−0.0573 − 0.0994i)23-s + (0.400 − 0.692i)25-s + (0.962 + 0.272i)27-s + (−0.919 + 1.59i)29-s + (0.668 + 1.15i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.359750 + 0.762283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.359750 + 0.762283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.724 - 1.25i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.72 - 2.98i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.94 - 3.37i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (0.275 + 0.476i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.94 - 8.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.72 - 6.45i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 + (-1.05 - 1.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.17 + 10.6i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.17 + 7.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.898T + 53T^{2} \) |
| 59 | \( 1 + (0.174 + 0.301i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.949 - 1.64i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.17 + 2.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 + (-4.27 + 7.40i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.72 - 4.71i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + (2.94 - 5.10i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.09979827840358063532059285352, −10.94486538209500303594410782222, −10.48888636665383007397842854825, −9.326634800223830272000323088991, −8.696995046746005902706375761896, −6.94507973022445647786709110214, −6.24416714420139548338665191609, −4.99058025842595815179822458880, −4.01275612175959312449128796544, −2.39291616219887072688479580107,
0.66540860896531064584168366888, 2.52742618224521092967301665505, 4.30408810415546339241063490909, 5.73263954580508246532527056950, 6.30738659467173641567699437648, 7.66031989695406549814228213055, 8.356799357655653873601405921314, 9.620570169482134375546961729178, 10.99081201531759967228128504234, 11.16532499897366687412361124976