| L(s) = 1 | + (−1.03 − 0.964i)2-s + (−0.759 + 1.55i)3-s + (0.140 + 1.99i)4-s + (−0.857 − 2.35i)5-s + (2.28 − 0.877i)6-s + (0.442 + 2.51i)7-s + (1.77 − 2.19i)8-s + (−1.84 − 2.36i)9-s + (−1.38 + 3.26i)10-s + (−1.32 + 3.65i)11-s + (−3.21 − 1.29i)12-s + (−1.81 + 2.15i)13-s + (1.96 − 3.02i)14-s + (4.31 + 0.454i)15-s + (−3.96 + 0.559i)16-s + (−3.79 + 6.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.731 − 0.681i)2-s + (−0.438 + 0.898i)3-s + (0.0701 + 0.997i)4-s + (−0.383 − 1.05i)5-s + (0.933 − 0.358i)6-s + (0.167 + 0.949i)7-s + (0.628 − 0.777i)8-s + (−0.615 − 0.788i)9-s + (−0.437 + 1.03i)10-s + (−0.400 + 1.10i)11-s + (−0.927 − 0.374i)12-s + (−0.502 + 0.598i)13-s + (0.524 − 0.808i)14-s + (1.11 + 0.117i)15-s + (−0.990 + 0.139i)16-s + (−0.919 + 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.246420 + 0.328979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246420 + 0.328979i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.03 + 0.964i)T \) |
| 3 | \( 1 + (0.759 - 1.55i)T \) |
| good | 5 | \( 1 + (0.857 + 2.35i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.442 - 2.51i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.32 - 3.65i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.81 - 2.15i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.79 - 6.56i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.25 - 2.45i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.488 + 2.77i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.68 - 2.00i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.605 + 3.43i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-6.91 - 3.99i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.68 + 6.44i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.38 - 3.81i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.48 - 8.41i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 + (0.584 + 1.60i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.343 - 0.0605i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.45 + 1.72i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.53 - 4.38i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.34 - 2.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.652 - 0.547i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.59 - 6.67i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.02 - 5.24i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.71 - 1.71i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.43715118895108809722308296150, −11.61596395343308772804026077204, −10.58318786816182055248253507991, −9.711953057050204934298433388505, −8.766490207170365837457326093356, −8.228534637701709829009382239607, −6.45403443709391408936993179912, −4.82315782183034092139063675561, −4.13018533429503293230355057687, −2.14431162353658169508702732653,
0.44401681717611281343269714864, 2.70322551151998100730561665192, 4.94062560974497186201047094022, 6.27147744715845574714841613291, 7.13449637712811850987351950872, 7.64530892516033366286036732241, 8.780268507701866266898554929235, 10.35222152394780627539144777609, 10.96603186671730118071829818817, 11.61362177170730738705173485008
