L(s) = 1 | + (0.923 + 1.60i)2-s + (0.5 − 0.866i)3-s + (−1.20 + 2.09i)4-s + (0.382 + 0.662i)5-s + 1.84·6-s − 2.61·8-s + (−0.499 − 0.866i)9-s + (−0.707 + 1.22i)10-s + (−0.923 + 1.60i)11-s + (1.20 + 2.09i)12-s + 13-s + 0.765·15-s + (−1.20 − 2.09i)16-s + (0.923 − 1.60i)18-s − 1.84·20-s + ⋯ |
L(s) = 1 | + (0.923 + 1.60i)2-s + (0.5 − 0.866i)3-s + (−1.20 + 2.09i)4-s + (0.382 + 0.662i)5-s + 1.84·6-s − 2.61·8-s + (−0.499 − 0.866i)9-s + (−0.707 + 1.22i)10-s + (−0.923 + 1.60i)11-s + (1.20 + 2.09i)12-s + 13-s + 0.765·15-s + (−1.20 − 2.09i)16-s + (0.923 − 1.60i)18-s − 1.84·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.006554784\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.006554784\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 0.765T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 0.765T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 1.84T + T^{2} \) |
| 89 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.358393622861179596466435536825, −8.425055267449410899962366496132, −7.85995108464957084663347330956, −7.12635417464092388007879458039, −6.65804667436695777240921455197, −5.94760564789826142087262741312, −5.09096673522396321397163310694, −4.09966370189939021329753553559, −3.11467725275536631628873434230, −2.12330110576906116948732826970,
1.12129193899777116031768703525, 2.44133281784734210018987054130, 3.22385409675230795547387846150, 3.88141161333921382167288357297, 4.82298442995589944979856443925, 5.50786661339275550396356191484, 6.03776857584904517753260068665, 7.967554352898383854772937299081, 8.791828867126964940161035747494, 9.213429140549792312383685481665