L(s) = 1 | + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.441 − 0.764i)5-s + (−0.5 − 0.866i)6-s + (0.369 − 2.61i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.441 − 0.764i)10-s + (−0.775 − 1.34i)11-s + (−0.5 − 0.866i)12-s + (2.13 − 2.90i)13-s + (0.369 − 2.61i)14-s + (−0.441 + 0.764i)15-s + 16-s − 7.17·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (−0.197 − 0.341i)5-s + (−0.204 − 0.353i)6-s + (0.139 − 0.990i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.139 − 0.241i)10-s + (−0.233 − 0.405i)11-s + (−0.144 − 0.249i)12-s + (0.591 − 0.805i)13-s + (0.0988 − 0.700i)14-s + (−0.113 + 0.197i)15-s + 0.250·16-s − 1.74·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0580 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0580 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24627 - 1.32088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24627 - 1.32088i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.369 + 2.61i)T \) |
| 13 | \( 1 + (-2.13 + 2.90i)T \) |
good | 5 | \( 1 + (0.441 + 0.764i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.775 + 1.34i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 7.17T + 17T^{2} \) |
| 19 | \( 1 + (2.37 - 4.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + (-3.87 + 6.71i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.330T + 37T^{2} \) |
| 41 | \( 1 + (3.02 - 5.24i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.35 - 5.81i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.976 - 1.69i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.74 - 11.6i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 61 | \( 1 + (-7.17 + 12.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.75 - 6.49i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.00 - 8.67i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.93 - 3.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.67 - 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + (-5.79 - 10.0i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.96866502458667119315711081590, −9.954597053034263186958299386865, −8.429769433594877043548579017820, −7.896102210224936528321946495970, −6.69200959027473877299692886377, −6.08388981409262132811927742313, −4.78792031049237203251871724987, −4.02767022098173651880493993052, −2.59630183762964731391340034771, −0.892733128753621187788149517683,
2.15184368730451333108424620849, 3.30973445519537675043719688581, 4.61785409446431762100936562007, 5.17246549299940839618912358782, 6.55272851635679152454052019227, 6.94558969087238909543736203307, 8.732645374438572317701982487113, 9.011691957154237023095661708918, 10.53029975130224415779456002874, 11.10746290326000497339070858240