| L(s) = 1 | + (1.14 − 1.30i)3-s + (−1.41 + 2.23i)7-s + (−0.381 − 2.97i)9-s + 1.13i·11-s − 0.333·13-s − 4.20i·17-s − 1.95i·19-s + (1.28 + 4.39i)21-s + 2.54·23-s + (−4.30 − 2.90i)27-s − 8.49i·29-s − 5.11i·31-s + (1.47 + 1.30i)33-s + 2.23i·37-s + (−0.381 + 0.434i)39-s + ⋯ |
| L(s) = 1 | + (0.660 − 0.750i)3-s + (−0.534 + 0.845i)7-s + (−0.127 − 0.991i)9-s + 0.342i·11-s − 0.0925·13-s − 1.02i·17-s − 0.448i·19-s + (0.281 + 0.959i)21-s + 0.529·23-s + (−0.828 − 0.559i)27-s − 1.57i·29-s − 0.918i·31-s + (0.257 + 0.226i)33-s + 0.367i·37-s + (−0.0611 + 0.0695i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.740381808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740381808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.14 + 1.30i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
| good | 11 | \( 1 - 1.13iT - 11T^{2} \) |
| 13 | \( 1 + 0.333T + 13T^{2} \) |
| 17 | \( 1 + 4.20iT - 17T^{2} \) |
| 19 | \( 1 + 1.95iT - 19T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 + 8.49iT - 29T^{2} \) |
| 31 | \( 1 + 5.11iT - 31T^{2} \) |
| 37 | \( 1 - 2.23iT - 37T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 + 0.527iT - 43T^{2} \) |
| 47 | \( 1 + 7.42iT - 47T^{2} \) |
| 53 | \( 1 - 8.22T + 53T^{2} \) |
| 59 | \( 1 + 3.59T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 6.70iT - 67T^{2} \) |
| 71 | \( 1 + 10.7iT - 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 - 5.81iT - 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 9.69T + 97T^{2} \) |
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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
−8.993589958207773418917024109784, −8.063428859522237780619136154243, −7.38354874083505125615716305720, −6.60771732101794050072810370966, −5.90708396866452897463437107487, −4.90314280614855711792539682394, −3.74763038630917389963367713884, −2.71617284501338297157051172208, −2.16841918407169936774808876932, −0.57492350888771121942729988801,
1.36961983446613691502164983657, 2.80059949683021740551113848038, 3.57738665315351325680965313847, 4.24990216936128653835285123833, 5.21256537981556653506055906960, 6.15709281762495469741784642054, 7.13678495016060904101261774086, 7.81261633620462316796235724274, 8.732440170638532639502960851936, 9.200484306606193183762667915304
