L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 − 0.5i)5-s + (0.5 + 0.866i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 1.5i)11-s + (0.499 − 0.866i)15-s + (0.866 + 0.499i)21-s + (0.499 − 0.866i)25-s − 0.999i·27-s − 1.73·29-s + (0.5 + 0.866i)31-s + (−1.5 − 0.866i)33-s + (0.866 + 0.499i)35-s − 0.999i·45-s + (−0.499 + 0.866i)49-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 − 0.5i)5-s + (0.5 + 0.866i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 1.5i)11-s + (0.499 − 0.866i)15-s + (0.866 + 0.499i)21-s + (0.499 − 0.866i)25-s − 0.999i·27-s − 1.73·29-s + (0.5 + 0.866i)31-s + (−1.5 − 0.866i)33-s + (0.866 + 0.499i)35-s − 0.999i·45-s + (−0.499 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.015461278\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.015461278\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 11 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 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 | \( 1 + 1.73T + T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−8.626838608420905529632864379369, −8.208262147347088806641447228895, −7.35566419338377136988310591336, −6.32519363628123051053998326278, −5.64864481010685982334220026183, −5.11261790111566541857994766914, −3.83607211641063197092203292660, −2.83129505915355814936910632496, −2.21405427560329911039044911435, −1.16033213190011212783963454945,
1.77898960763403525164835240763, 2.30613873186312757674385410765, 3.37828360226792300866770150108, 4.30255975086990826350758572686, 4.94419933224220499746627567745, 5.78445422622444924915063157382, 7.04061963329356884778400726219, 7.39751700102035606303451319933, 8.076853912668849779704105368485, 9.044651534568144376897532817426