L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.20 − 2.08i)3-s + (−0.499 − 0.866i)4-s + (1.10 − 1.91i)5-s + 2.40·6-s + (−1.36 − 2.26i)7-s + 0.999·8-s + (−1.39 + 2.41i)9-s + (1.10 + 1.91i)10-s + (0.593 + 1.02i)11-s + (−1.20 + 2.08i)12-s + 1.24·13-s + (2.64 − 0.0506i)14-s − 5.31·15-s + (−0.5 + 0.866i)16-s + (−2.98 − 5.16i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.694 − 1.20i)3-s + (−0.249 − 0.433i)4-s + (0.493 − 0.855i)5-s + 0.982·6-s + (−0.516 − 0.856i)7-s + 0.353·8-s + (−0.465 + 0.806i)9-s + (0.349 + 0.604i)10-s + (0.179 + 0.310i)11-s + (−0.347 + 0.601i)12-s + 0.344·13-s + (0.706 − 0.0135i)14-s − 1.37·15-s + (−0.125 + 0.216i)16-s + (−0.723 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.195623 - 0.625629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.195623 - 0.625629i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.36 + 2.26i)T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + (1.20 + 2.08i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.10 + 1.91i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.593 - 1.02i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 + (2.98 + 5.16i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.09 - 1.90i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.87 - 6.70i)T + (-11.5 - 19.9i)T^{2} \) |
| 31 | \( 1 + (4.65 + 8.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.45 + 5.97i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.840 + 1.45i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.98 - 6.90i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.05 - 7.02i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.07 + 10.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.61T + 71T^{2} \) |
| 73 | \( 1 + (0.0197 + 0.0341i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.21 + 2.09i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.46T + 83T^{2} \) |
| 89 | \( 1 + (-3.63 + 6.29i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.23T + 97T^{2} \) |
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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
−10.98009038121140638970365333484, −9.685982693455542405065185313829, −9.136307630811302790290857654993, −7.68838651775865706601745616671, −7.22769060341325600981622491448, −6.18305212691157326318461745036, −5.50918958540886189228286610358, −4.14110323070457783537543150769, −1.78696425079981958534911999372, −0.51679547235511003298218947746,
2.33358799045613421298476240414, 3.53749833900974091040042168261, 4.64405376846836677555993573994, 5.96806197474045243500397115719, 6.54959343283795841079532592175, 8.379028225815736979159912747438, 9.170967440793557598421622803482, 10.05764615170293195766018385002, 10.70330827786818053825528324202, 11.16357854572745766344532540822