| L(s) = 1 | + (−1.80 + 0.172i)2-s + (0.690 + 0.723i)3-s + (1.25 − 0.242i)4-s + (2.30 + 1.18i)5-s + (−1.36 − 1.18i)6-s + (2.63 + 0.200i)7-s + (1.24 − 0.366i)8-s + (−0.0475 + 0.998i)9-s + (−4.36 − 1.74i)10-s + (−0.336 + 3.52i)11-s + (1.04 + 0.743i)12-s + (3.67 − 0.528i)13-s + (−4.79 + 0.0918i)14-s + (0.730 + 2.48i)15-s + (−4.56 + 1.82i)16-s + (−1.78 − 5.14i)17-s + ⋯ |
| L(s) = 1 | + (−1.27 + 0.121i)2-s + (0.398 + 0.417i)3-s + (0.629 − 0.121i)4-s + (1.03 + 0.531i)5-s + (−0.558 − 0.484i)6-s + (0.997 + 0.0759i)7-s + (0.441 − 0.129i)8-s + (−0.0158 + 0.332i)9-s + (−1.37 − 0.552i)10-s + (−0.101 + 1.06i)11-s + (0.301 + 0.214i)12-s + (1.01 − 0.146i)13-s + (−1.28 + 0.0245i)14-s + (0.188 + 0.642i)15-s + (−1.14 + 0.457i)16-s + (−0.432 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00174 + 0.522548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00174 + 0.522548i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.690 - 0.723i)T \) |
| 7 | \( 1 + (-2.63 - 0.200i)T \) |
| 23 | \( 1 + (-3.96 - 2.70i)T \) |
| good | 2 | \( 1 + (1.80 - 0.172i)T + (1.96 - 0.378i)T^{2} \) |
| 5 | \( 1 + (-2.30 - 1.18i)T + (2.90 + 4.07i)T^{2} \) |
| 11 | \( 1 + (0.336 - 3.52i)T + (-10.8 - 2.08i)T^{2} \) |
| 13 | \( 1 + (-3.67 + 0.528i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (1.78 + 5.14i)T + (-13.3 + 10.5i)T^{2} \) |
| 19 | \( 1 + (-2.68 + 7.76i)T + (-14.9 - 11.7i)T^{2} \) |
| 29 | \( 1 + (5.10 - 5.89i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (6.35 + 1.54i)T + (27.5 + 14.2i)T^{2} \) |
| 37 | \( 1 + (-1.18 - 0.0566i)T + (36.8 + 3.51i)T^{2} \) |
| 41 | \( 1 + (-2.75 - 4.29i)T + (-17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.88 + 6.40i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (0.107 - 0.0620i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.63 - 8.43i)T + (-12.4 - 51.5i)T^{2} \) |
| 59 | \( 1 + (0.888 - 2.22i)T + (-42.7 - 40.7i)T^{2} \) |
| 61 | \( 1 + (-4.92 - 4.69i)T + (2.90 + 60.9i)T^{2} \) |
| 67 | \( 1 + (7.92 - 5.64i)T + (21.9 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-0.643 + 1.40i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.56 - 8.12i)T + (-67.7 + 27.1i)T^{2} \) |
| 79 | \( 1 + (0.233 + 0.296i)T + (-18.6 + 76.7i)T^{2} \) |
| 83 | \( 1 + (11.7 + 7.57i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (3.92 + 16.1i)T + (-79.1 + 40.7i)T^{2} \) |
| 97 | \( 1 + (-4.66 + 2.99i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.97773916212596116476353921515, −9.985879906250322582144754182214, −9.155706536767795047473271647227, −8.896089950020968188339673458031, −7.43299644431387647218658263769, −7.09007344832462623786600555151, −5.44889873054687175501531667231, −4.50258763886775047077214442393, −2.68124320680034697705001019093, −1.52809377162805686762632030530,
1.26636305751458739947012121941, 1.91840538952144126629054140110, 3.85420110403822294134062860069, 5.42830094878005895482464069454, 6.25565397277303200748915707524, 7.75951973220429393272195283280, 8.303823512729494423115328565913, 8.936020026068887565810569130945, 9.728291504800804727470148167415, 10.80502614822642122297759638504
