L(s) = 1 | + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.5 + 0.866i)6-s + (−1 − 1.73i)7-s + 0.999·8-s − 2·9-s + (0.5 − 0.866i)10-s + (−0.499 − 0.866i)12-s + (−1 + 1.73i)13-s + 1.99·14-s − 15-s + (−0.5 + 0.866i)16-s + (−3.22 + 5.58i)17-s + (1 − 1.73i)18-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + 0.577·3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (−0.204 + 0.353i)6-s + (−0.377 − 0.654i)7-s + 0.353·8-s − 0.666·9-s + (0.158 − 0.273i)10-s + (−0.144 − 0.249i)12-s + (−0.277 + 0.480i)13-s + 0.534·14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.781 + 1.35i)17-s + (0.235 − 0.408i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0137455 + 0.394673i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0137455 + 0.394673i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (0.276 + 8.18i)T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.22 - 7.31i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.72 - 6.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.72 - 4.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.72 + 8.18i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 9.44T + 53T^{2} \) |
| 59 | \( 1 + 0.446T + 59T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (4.72 + 8.18i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.776 - 1.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.72 - 2.98i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + (-2.22 + 3.85i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.57280199806475869006098669777, −10.11418750109050764453945261591, −8.824343891616171749115835055253, −8.432478588935513944570656492015, −7.55937840411831058354189560162, −6.59771370604837449373916304024, −5.80057318961531591949219658152, −4.32120361969486359156557723068, −3.56999249127460899495311542879, −1.93594985545637316623532541233,
0.20531021329485706268201773127, 2.47388025158436851477249216373, 2.92873602934317063659094526084, 4.30215732299665809351944490528, 5.38760884581231267846360778791, 6.73922398861450374116841939217, 7.65163665459918393504279968598, 8.778780067713602355956133973687, 8.977823484082973185871111031417, 9.960920219766353852121068327175