L(s) = 1 | + (−0.5 + 0.866i)5-s + (−1 − 1.73i)7-s + (−1 + 1.73i)13-s − 3·17-s + 5·19-s + (−1.5 + 2.59i)23-s + (−0.499 − 0.866i)25-s + (3 + 5.19i)29-s + (−2.5 + 4.33i)31-s + 1.99·35-s + 2·37-s + (−6 + 10.3i)41-s + (−4 − 6.92i)43-s + (6 + 10.3i)47-s + (1.50 − 2.59i)49-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (−0.377 − 0.654i)7-s + (−0.277 + 0.480i)13-s − 0.727·17-s + 1.14·19-s + (−0.312 + 0.541i)23-s + (−0.0999 − 0.173i)25-s + (0.557 + 0.964i)29-s + (−0.449 + 0.777i)31-s + 0.338·35-s + 0.328·37-s + (−0.937 + 1.62i)41-s + (−0.609 − 1.05i)43-s + (0.875 + 1.51i)47-s + (0.214 − 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9959256481\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9959256481\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 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 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 16T + 73T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.658046576257505396066782608257, −8.894118974858806846509640377690, −7.922361300288781335385497222834, −7.08543005360257846533748695637, −6.66366405066513570604633639399, −5.51398481501151096020107827863, −4.56164670331893563799858427284, −3.63834738003612573374401987901, −2.77877919166242687809943407586, −1.33428248650444441071186419701,
0.39719415030826636888928434503, 2.06303190498852804834951384172, 3.07280062367362272739357168848, 4.13392715776059607618726701998, 5.11112619081516648838162756916, 5.84411902691293804897692935056, 6.74483944259291106520329136216, 7.67967462989094291475036555323, 8.405355951009289857382128446472, 9.188121844361043554238643821877