| L(s) = 1 | + (−0.670 + 1.24i)2-s + (1.16 + 1.24i)3-s + (−1.09 − 1.67i)4-s + (−0.922 − 2.03i)5-s + (−2.32 + 0.614i)6-s + (−2.59 − 0.514i)7-s + (2.81 − 0.248i)8-s + (0.00726 − 0.110i)9-s + (3.15 + 0.217i)10-s + (0.0927 + 0.149i)11-s + (0.794 − 3.30i)12-s + (0.575 + 5.84i)13-s + (2.38 − 2.88i)14-s + (1.45 − 3.50i)15-s + (−1.58 + 3.67i)16-s + (1.81 + 0.238i)17-s + ⋯ |
| L(s) = 1 | + (−0.474 + 0.880i)2-s + (0.670 + 0.716i)3-s + (−0.549 − 0.835i)4-s + (−0.412 − 0.910i)5-s + (−0.948 + 0.250i)6-s + (−0.980 − 0.194i)7-s + (0.996 − 0.0879i)8-s + (0.00242 − 0.0369i)9-s + (0.997 + 0.0687i)10-s + (0.0279 + 0.0449i)11-s + (0.229 − 0.954i)12-s + (0.159 + 1.62i)13-s + (0.636 − 0.771i)14-s + (0.375 − 0.906i)15-s + (−0.395 + 0.918i)16-s + (0.439 + 0.0578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.350679 + 0.916880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.350679 + 0.916880i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.670 - 1.24i)T \) |
| 7 | \( 1 + (2.59 + 0.514i)T \) |
| good | 3 | \( 1 + (-1.16 - 1.24i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (0.922 + 2.03i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (-0.0927 - 0.149i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (-0.575 - 5.84i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-1.81 - 0.238i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (0.984 - 5.96i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (-2.21 - 2.52i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-4.61 - 8.63i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (10.3 - 2.76i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (0.0775 - 0.206i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (-1.42 - 7.18i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.314 + 0.0952i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (2.05 - 2.68i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (0.176 + 0.284i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (-0.0518 + 0.0723i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (-4.06 + 0.948i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (-2.40 - 2.56i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (2.75 + 4.11i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-14.1 + 6.98i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (1.33 + 10.1i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (5.30 - 6.46i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (-2.63 - 7.75i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (2.29 - 2.29i)T - 97iT^{2} \) |
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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
−9.986631047178576459538481693375, −9.306455659530536724223949181855, −8.921960874766401245051265647764, −8.134915208549954108839150257677, −7.04687230136303169556815689522, −6.31671058156981840751978182979, −5.12940121512075118796683047528, −4.18703005690905882106263543234, −3.48384522845630608563972651450, −1.38502293721308910395250610849,
0.54720309847503886536414019638, 2.45100367247110133171138368296, 2.91440279923402879283005384075, 3.80016756943616366166832468023, 5.37093402325070410829487743601, 6.77890982861732287469782136300, 7.41248397174627152519905770062, 8.159483439476771282149139872251, 8.946927184168368409603002988218, 9.881901513986290384125699396675
