L(s) = 1 | + (−0.214 + 0.156i)2-s + (0.309 + 0.951i)3-s + (−0.596 + 1.83i)4-s + (−3.10 − 2.25i)5-s + (−0.214 − 0.156i)6-s + (−0.868 + 2.67i)7-s + (−0.322 − 0.992i)8-s + (−0.809 + 0.587i)9-s + 1.02·10-s + (1.88 − 2.72i)11-s − 1.92·12-s + (0.809 − 0.587i)13-s + (−0.230 − 0.709i)14-s + (1.18 − 3.65i)15-s + (−2.89 − 2.10i)16-s + (−2.81 − 2.04i)17-s + ⋯ |
L(s) = 1 | + (−0.151 + 0.110i)2-s + (0.178 + 0.549i)3-s + (−0.298 + 0.917i)4-s + (−1.39 − 1.01i)5-s + (−0.0877 − 0.0637i)6-s + (−0.328 + 1.01i)7-s + (−0.114 − 0.350i)8-s + (−0.269 + 0.195i)9-s + 0.322·10-s + (0.568 − 0.822i)11-s − 0.556·12-s + (0.224 − 0.163i)13-s + (−0.0616 − 0.189i)14-s + (0.306 − 0.943i)15-s + (−0.724 − 0.526i)16-s + (−0.683 − 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0297937 - 0.0584969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0297937 - 0.0584969i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-1.88 + 2.72i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
good | 2 | \( 1 + (0.214 - 0.156i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (3.10 + 2.25i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.868 - 2.67i)T + (-5.66 - 4.11i)T^{2} \) |
| 17 | \( 1 + (2.81 + 2.04i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.45 + 4.46i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 8.27T + 23T^{2} \) |
| 29 | \( 1 + (-0.784 + 2.41i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.85 - 2.80i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.99 + 9.21i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.26 - 10.0i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.76T + 43T^{2} \) |
| 47 | \( 1 + (-1.73 - 5.32i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (8.20 - 5.96i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.599 + 1.84i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.435 + 0.316i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 0.204T + 67T^{2} \) |
| 71 | \( 1 + (7.34 + 5.33i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.24 - 13.0i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (10.2 - 7.45i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (14.3 + 10.4i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + (-11.3 + 8.24i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.25146922506259775534218500496, −9.478748308874694516098649211659, −8.831218298447388055948704298577, −8.396304264083528417222811622256, −7.49261905401711899657983830573, −6.05050228496216823159329364670, −4.65505121993265933179576371254, −3.97706333201726809780010599614, −2.91349398634755514518844833141, −0.04228037465564830460371285573,
1.84389601754726470907132064098, 3.68319754315846823158876071599, 4.31749232137689014381068377998, 6.18602246643933087024406927596, 6.86283005622930703560067614300, 7.70342044769549105829466373870, 8.623758372044833113060166191505, 9.998375476428226311501355435728, 10.47878115621962913460720505456, 11.43510809688004351483419126055