L(s) = 1 | + (−0.621 + 1.27i)2-s + (1.72 − 0.121i)3-s + (−1.22 − 1.57i)4-s + (1 + 2i)5-s + (−0.919 + 2.27i)6-s + 7-s + (2.76 − 0.578i)8-s + (2.97 − 0.419i)9-s + (−3.16 + 0.0276i)10-s + 3.45·11-s + (−2.31 − 2.57i)12-s − 4.83i·13-s + (−0.621 + 1.27i)14-s + (1.97 + 3.33i)15-s + (−0.985 + 3.87i)16-s − 5.94·17-s + ⋯ |
L(s) = 1 | + (−0.439 + 0.898i)2-s + (0.997 − 0.0700i)3-s + (−0.613 − 0.789i)4-s + (0.447 + 0.894i)5-s + (−0.375 + 0.926i)6-s + 0.377·7-s + (0.978 − 0.204i)8-s + (0.990 − 0.139i)9-s + (−0.999 + 0.00874i)10-s + 1.04·11-s + (−0.667 − 0.744i)12-s − 1.34i·13-s + (−0.166 + 0.339i)14-s + (0.508 + 0.860i)15-s + (−0.246 + 0.969i)16-s − 1.44·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39004 + 0.945651i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39004 + 0.945651i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.621 - 1.27i)T \) |
| 3 | \( 1 + (-1.72 + 0.121i)T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 3.45T + 11T^{2} \) |
| 13 | \( 1 + 4.83iT - 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 19 | \( 1 - 1.08iT - 19T^{2} \) |
| 23 | \( 1 + 0.596iT - 23T^{2} \) |
| 29 | \( 1 - 4.83iT - 29T^{2} \) |
| 31 | \( 1 - 9.56iT - 31T^{2} \) |
| 37 | \( 1 + 2.91iT - 37T^{2} \) |
| 41 | \( 1 + 6.91iT - 41T^{2} \) |
| 43 | \( 1 - 7.39T + 43T^{2} \) |
| 47 | \( 1 - 0.242iT - 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 3.25T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 5.71T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 0.949iT - 79T^{2} \) |
| 83 | \( 1 + 16.9iT - 83T^{2} \) |
| 89 | \( 1 + 5.23iT - 89T^{2} \) |
| 97 | \( 1 - 3.30iT - 97T^{2} \) |
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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
−10.84532164760799796450634542704, −10.37647187539469739660922524985, −9.208707060766649844772330213310, −8.714604647195644652880670203631, −7.59237888891814503443928691959, −6.91602830983979071661029809374, −5.98655374117242715846154090888, −4.59884925978968579407595531668, −3.25991482964941801418624682659, −1.70651328880567396191403880815,
1.48702576703692628709898045689, 2.39756529951137475425605025371, 4.18175171089377403794672115766, 4.47966605557148339590012881237, 6.47498175631935730780652735514, 7.75015849915149071157319870093, 8.671113014564968156702217359399, 9.297179360771287514173378340793, 9.688149118156443203728150855657, 11.12032017962887595124076607093