L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.499i)6-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s − 0.999i·12-s + (−0.5 + 0.866i)16-s − 1.73·17-s − 0.999·18-s + 1.73·19-s + (−0.499 − 0.866i)22-s + (0.866 + 0.5i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.499i)6-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s − 0.999i·12-s + (−0.5 + 0.866i)16-s − 1.73·17-s − 0.999·18-s + 1.73·19-s + (−0.499 − 0.866i)22-s + (0.866 + 0.5i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.128393517\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128393517\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 19 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.73T + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.085100746353997677037460231511, −8.211623799069015982041501846470, −7.64059447093327391591002648970, −7.08642865150607511677901833763, −6.20208793300539520749907447819, −5.05920633259048110981569120012, −4.73035543634894540203255999174, −3.71835541975936937054137983387, −2.56839747743186034436499420823, −1.56910118947029171975943124022,
0.70994332287186614771724499622, 1.99808935674086793156071730533, 2.68603060676965850021642576473, 3.50462363960872984512341434575, 4.25714491406107123714254798086, 5.33669425285980143677283930936, 6.45904560032753255785622720586, 7.31416730295540408200044812637, 7.88112191686739295363481958541, 8.626548594513268137497645627391