L(s) = 1 | + (1.68 − 2.92i)5-s + (−2.35 − 4.07i)7-s + (−0.437 − 0.758i)11-s + (−0.686 + 1.18i)13-s + 2.37·17-s − 5.57·19-s + (2.35 − 4.07i)23-s + (−3.18 − 5.51i)25-s + (2.68 + 4.65i)29-s + (3.22 − 5.58i)31-s − 15.8·35-s − 4·37-s + (0.5 − 0.866i)41-s + (−0.437 − 0.758i)43-s + (2.35 + 4.07i)47-s + ⋯ |
L(s) = 1 | + (0.754 − 1.30i)5-s + (−0.888 − 1.53i)7-s + (−0.131 − 0.228i)11-s + (−0.190 + 0.329i)13-s + 0.575·17-s − 1.27·19-s + (0.490 − 0.849i)23-s + (−0.637 − 1.10i)25-s + (0.498 + 0.863i)29-s + (0.579 − 1.00i)31-s − 2.68·35-s − 0.657·37-s + (0.0780 − 0.135i)41-s + (−0.0667 − 0.115i)43-s + (0.342 + 0.594i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.241885134\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.241885134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.68 + 2.92i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.35 + 4.07i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.437 + 0.758i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.686 - 1.18i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 + 5.57T + 19T^{2} \) |
| 23 | \( 1 + (-2.35 + 4.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.68 - 4.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.22 + 5.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.437 + 0.758i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.35 - 4.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (4.26 - 7.38i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.05 + 1.83i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.26 - 7.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + (3.22 + 5.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.47 - 2.55i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.992124230675252700433490822366, −8.335615805394894912378452179158, −7.31131647193467126495537783647, −6.53430952672797024131354339739, −5.77534928868979792035759869397, −4.65943742904743401218562542840, −4.15985072936494514591251257995, −2.91996682471425819857578078915, −1.48761920906037437307952990886, −0.45407993916457973362391304019,
2.00718647677469567645625053038, 2.76648596671658269251234551829, 3.40572949544772737927450904017, 4.98053571269671431570479173686, 5.88256622681163220268964631898, 6.35382743412193742830372601289, 7.06936546389434338095652904801, 8.170445229203614293427854780242, 9.017471036219981850743453912827, 9.770870192499216616007600773369