| L(s) = 1 | + (4.69 + 8.13i)7-s + (−15.4 + 8.90i)11-s + (−6.92 + 11.9i)13-s + 30.7i·17-s + 28.5·19-s + (−10.6 − 6.16i)23-s + (5.02 − 2.90i)29-s + (−3.25 + 5.63i)31-s − 66.5·37-s + (33.0 + 19.0i)41-s + (27.5 + 47.7i)43-s + (14.2 − 8.23i)47-s + (−19.5 + 33.9i)49-s − 69.8i·53-s + (−91.2 − 52.6i)59-s + ⋯ |
| L(s) = 1 | + (0.670 + 1.16i)7-s + (−1.40 + 0.809i)11-s + (−0.532 + 0.922i)13-s + 1.80i·17-s + 1.50·19-s + (−0.464 − 0.267i)23-s + (0.173 − 0.100i)29-s + (−0.104 + 0.181i)31-s − 1.79·37-s + (0.805 + 0.464i)41-s + (0.641 + 1.11i)43-s + (0.303 − 0.175i)47-s + (−0.399 + 0.692i)49-s − 1.31i·53-s + (−1.54 − 0.893i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.052140585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.052140585\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-4.69 - 8.13i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (15.4 - 8.90i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (6.92 - 11.9i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 30.7iT - 289T^{2} \) |
| 19 | \( 1 - 28.5T + 361T^{2} \) |
| 23 | \( 1 + (10.6 + 6.16i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-5.02 + 2.90i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (3.25 - 5.63i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 66.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-33.0 - 19.0i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-27.5 - 47.7i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-14.2 + 8.23i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 69.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (91.2 + 52.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.5 + 58.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-22.9 + 39.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 31.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 73.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + (47.3 + 81.9i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-13.4 + 7.73i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 52.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-38.1 - 66.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.069968931271172517150736447155, −8.114054839043716455648417435607, −7.80843693717725245820879531095, −6.77860844918900229609295534822, −5.85358229145250395252478858005, −5.14418629510190570137110280455, −4.56094912721722866956989190061, −3.33459450964991957304272456443, −2.24767877943192978054764010431, −1.69312889433382225676112303341,
0.25826054670561432732544102810, 1.05862385047669554620892295502, 2.61168004415321547496458082170, 3.24246299947131416184558793596, 4.38170012160990416458504240763, 5.33376330108093478937751504517, 5.54956530484924658182176093698, 7.17277129107524416101805196655, 7.46657205752253487228159532031, 8.009435486650557775258486869339
