L(s) = 1 | + (0.866 − 1.5i)3-s + (1.73 + 2i)7-s + (−1.5 − 2.59i)9-s − 5.19i·11-s + 1.73·13-s − 3i·17-s − 3.46i·19-s + (4.5 − 0.866i)21-s − 5.19·27-s + 5.19i·29-s + 10.3i·31-s + (−7.79 − 4.5i)33-s − 8i·37-s + (1.49 − 2.59i)39-s − 6·41-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)3-s + (0.654 + 0.755i)7-s + (−0.5 − 0.866i)9-s − 1.56i·11-s + 0.480·13-s − 0.727i·17-s − 0.794i·19-s + (0.981 − 0.188i)21-s − 1.00·27-s + 0.964i·29-s + 1.86i·31-s + (−1.35 − 0.783i)33-s − 1.31i·37-s + (0.240 − 0.416i)39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.072830687\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.072830687\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.19iT - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−8.737676475813404816037026095886, −8.347428049674227334076636718864, −7.29106596081460710470243004783, −6.66265139196969711554862606606, −5.67533172264170911991100913524, −5.12576901823567802843794299595, −3.60667396462845503328536889680, −2.91863239066154896975224827689, −1.87953107668406121444750877928, −0.69768591876391791731727586920,
1.53282164098547688084283950687, 2.52122790369743667442264449462, 3.94609370429257011483942276148, 4.21041796029086727501978295933, 5.11713253985701215545896092351, 6.11901192788935026134681091455, 7.18707490707429801558711060333, 8.007438785864637452580012289619, 8.362209909617148918505047260520, 9.648182843569644839764575895431