L(s) = 1 | + i·2-s + (−0.5 − 1.65i)3-s − 4-s + (1.91 − 1.15i)5-s + (1.65 − 0.5i)6-s + 3.21i·7-s − i·8-s + (−2.5 + 1.65i)9-s + (1.15 + 1.91i)10-s + 4.31i·11-s + (0.5 + 1.65i)12-s + 3.31·13-s − 3.21·14-s + (−2.87 − 2.59i)15-s + 16-s + 3.21·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.288 − 0.957i)3-s − 0.5·4-s + (0.855 − 0.518i)5-s + (0.677 − 0.204i)6-s + 1.21i·7-s − 0.353i·8-s + (−0.833 + 0.552i)9-s + (0.366 + 0.604i)10-s + 1.30i·11-s + (0.144 + 0.478i)12-s + 0.919·13-s − 0.860·14-s + (−0.742 − 0.669i)15-s + 0.250·16-s + 0.780·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37366 + 0.499860i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37366 + 0.499860i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.5 + 1.65i)T \) |
| 5 | \( 1 + (-1.91 + 1.15i)T \) |
| 19 | \( 1 + (4.31 + 0.605i)T \) |
good | 7 | \( 1 - 3.21iT - 7T^{2} \) |
| 11 | \( 1 - 4.31iT - 11T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 - 8.25T + 29T^{2} \) |
| 31 | \( 1 + 2.61iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 3.82iT - 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 1.81iT - 73T^{2} \) |
| 79 | \( 1 - 15.3iT - 79T^{2} \) |
| 83 | \( 1 + 6.43T + 83T^{2} \) |
| 89 | \( 1 - 2.61T + 89T^{2} \) |
| 97 | \( 1 + 3.36T + 97T^{2} \) |
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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
−10.81024435039304268921233982717, −9.727366095426962241855385281793, −8.774360322781266351555593450392, −8.310610177631154624861490462099, −7.02381065053566138576728950320, −6.30462141542472313279751005870, −5.51369817387461804200516101872, −4.73958553200189097475567714687, −2.65190830678613713680380445554, −1.43489400938269139052240448467,
1.03292326427899549499258678015, 3.04632443277903002603449380817, 3.68887578599870942221811668942, 4.90734183302549374764122157878, 5.92591415126299207618053038508, 6.79898497328661438223456341823, 8.400901340166381553949305063903, 9.058871093426804019965927558579, 10.22173307659288407374285577723, 10.61040898467657513258625814788