| L(s) = 1 | + (1 − i)3-s + (−7 − 7i)7-s + 7i·9-s − 10·11-s + (−9 + 9i)13-s + (−1 − i)17-s + 8i·19-s − 14·21-s + (−23 + 23i)23-s + (16 + 16i)27-s − 8i·29-s + 14·31-s + (−10 + 10i)33-s + (−33 − 33i)37-s + 18i·39-s + ⋯ |
| L(s) = 1 | + (0.333 − 0.333i)3-s + (−1 − i)7-s + 0.777i·9-s − 0.909·11-s + (−0.692 + 0.692i)13-s + (−0.0588 − 0.0588i)17-s + 0.421i·19-s − 0.666·21-s + (−1 + i)23-s + (0.592 + 0.592i)27-s − 0.275i·29-s + 0.451·31-s + (−0.303 + 0.303i)33-s + (−0.891 − 0.891i)37-s + 0.461i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0512526 + 0.180416i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0512526 + 0.180416i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-1 + i)T - 9iT^{2} \) |
| 7 | \( 1 + (7 + 7i)T + 49iT^{2} \) |
| 11 | \( 1 + 10T + 121T^{2} \) |
| 13 | \( 1 + (9 - 9i)T - 169iT^{2} \) |
| 17 | \( 1 + (1 + i)T + 289iT^{2} \) |
| 19 | \( 1 - 8iT - 361T^{2} \) |
| 23 | \( 1 + (23 - 23i)T - 529iT^{2} \) |
| 29 | \( 1 + 8iT - 841T^{2} \) |
| 31 | \( 1 - 14T + 961T^{2} \) |
| 37 | \( 1 + (33 + 33i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 14T + 1.68e3T^{2} \) |
| 43 | \( 1 + (15 - 15i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (39 + 39i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-7 + 7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 56iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 42T + 3.72e3T^{2} \) |
| 67 | \( 1 + (7 + 7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 98T + 5.04e3T^{2} \) |
| 73 | \( 1 + (49 - 49i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 96iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (63 - 63i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 112iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (33 + 33i)T + 9.40e3iT^{2} \) |
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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
−11.41499513159046605609880188291, −10.22727968251262811586009858664, −9.924881399325737445108897709911, −8.572085299272496551984124415464, −7.54623056790744008073582091156, −7.03152782655453350323850298444, −5.72744098947145072525660119313, −4.48457765401652588862314413602, −3.26555131103176463657106291736, −1.96140755417840218347593722656,
0.07153432332835254383280210600, 2.53956892720900427113331112727, 3.30990526953465270386917161561, 4.80020209566613135711326567837, 5.90458906093560965323063219196, 6.77842899970267672950284916551, 8.120787698760519666875659126330, 8.896855359243206894741192457438, 9.848243468833479780588358365579, 10.34075574639242689201889643153
