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
−9.960920219766353852121068327175, −8.977823484082973185871111031417, −8.778780067713602355956133973687, −7.65163665459918393504279968598, −6.73922398861450374116841939217, −5.38760884581231267846360778791, −4.30215732299665809351944490528, −2.92873602934317063659094526084, −2.47388025158436851477249216373, −0.20531021329485706268201773127,
1.93594985545637316623532541233, 3.56999249127460899495311542879, 4.32120361969486359156557723068, 5.80057318961531591949219658152, 6.59771370604837449373916304024, 7.55937840411831058354189560162, 8.432478588935513944570656492015, 8.824343891616171749115835055253, 10.11418750109050764453945261591, 10.57280199806475869006098669777