L(s) = 1 | + (0.5 − 0.866i)2-s + (0.658 − 1.60i)3-s + (−0.499 − 0.866i)4-s + (1.00 + 0.578i)5-s + (−1.05 − 1.37i)6-s + (0.296 − 2.62i)7-s − 0.999·8-s + (−2.13 − 2.11i)9-s + (1.00 − 0.578i)10-s + (1.85 + 3.21i)11-s + (−1.71 + 0.230i)12-s + (0.361 − 3.58i)13-s + (−2.12 − 1.57i)14-s + (1.58 − 1.22i)15-s + (−0.5 + 0.866i)16-s + (−0.0128 − 0.0221i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.380 − 0.924i)3-s + (−0.249 − 0.433i)4-s + (0.448 + 0.258i)5-s + (−0.431 − 0.559i)6-s + (0.111 − 0.993i)7-s − 0.353·8-s + (−0.710 − 0.703i)9-s + (0.316 − 0.182i)10-s + (0.559 + 0.969i)11-s + (−0.495 + 0.0664i)12-s + (0.100 − 0.994i)13-s + (−0.568 − 0.419i)14-s + (0.409 − 0.315i)15-s + (−0.125 + 0.216i)16-s + (−0.00310 − 0.00537i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.810945 - 1.81341i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.810945 - 1.81341i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.658 + 1.60i)T \) |
| 7 | \( 1 + (-0.296 + 2.62i)T \) |
| 13 | \( 1 + (-0.361 + 3.58i)T \) |
good | 5 | \( 1 + (-1.00 - 0.578i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.85 - 3.21i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.0128 + 0.0221i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.122 + 0.211i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 + 1.36i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.51iT - 29T^{2} \) |
| 31 | \( 1 + (-2.24 - 3.89i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.29 - 4.79i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.471iT - 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 + (-0.203 - 0.117i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.44 + 3.72i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.80 + 1.61i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.2 - 5.91i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.24 - 3.60i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + (-0.784 - 1.35i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.01 + 5.21i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.64iT - 83T^{2} \) |
| 89 | \( 1 + (-8.11 - 4.68i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−10.36315110570496989723320468554, −9.909806457850190294186609419466, −8.697433274642466477438322072071, −7.70425262394725432235752949499, −6.83819794049396717073976867203, −5.98703879232188086947086479028, −4.62414506422921486582414706299, −3.46220715571455332109557445989, −2.29669621931408674924810930108, −1.06712027029137182699250967197,
2.27078858306348081264594677254, 3.60657899941217805975992294247, 4.55529920445653124811161509352, 5.69627277949044709831608536771, 6.14520197789995969292064192467, 7.69959027829538092565990846767, 8.695455006873448467708400069265, 9.156343783138950834837649632732, 9.924996091066767789091513140561, 11.37127823580717110593910039430