L(s) = 1 | + (0.519 − 0.299i)2-s + (0.801 + 1.38i)3-s + (−0.820 + 1.42i)4-s − 0.195i·5-s + (0.832 + 0.480i)6-s + (2.33 + 1.34i)7-s + 2.18i·8-s + (0.214 − 0.371i)9-s + (−0.0586 − 0.101i)10-s + (0.499 − 0.288i)11-s − 2.63·12-s + (−3.39 + 1.21i)13-s + 1.61·14-s + (0.271 − 0.157i)15-s + (−0.986 − 1.70i)16-s + (−1.81 + 3.14i)17-s + ⋯ |
L(s) = 1 | + (0.367 − 0.211i)2-s + (0.462 + 0.801i)3-s + (−0.410 + 0.710i)4-s − 0.0875i·5-s + (0.339 + 0.196i)6-s + (0.883 + 0.510i)7-s + 0.771i·8-s + (0.0714 − 0.123i)9-s + (−0.0185 − 0.0321i)10-s + (0.150 − 0.0869i)11-s − 0.759·12-s + (−0.941 + 0.337i)13-s + 0.432·14-s + (0.0702 − 0.0405i)15-s + (−0.246 − 0.427i)16-s + (−0.439 + 0.761i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39646 + 1.15152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39646 + 1.15152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (3.39 - 1.21i)T \) |
| 31 | \( 1 + iT \) |
good | 2 | \( 1 + (-0.519 + 0.299i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.801 - 1.38i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 0.195iT - 5T^{2} \) |
| 7 | \( 1 + (-2.33 - 1.34i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.499 + 0.288i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.81 - 3.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.74 + 1.01i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.32 - 2.28i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.15 - 2.00i)T + (-14.5 + 25.1i)T^{2} \) |
| 37 | \( 1 + (-6.65 + 3.84i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.10 - 4.10i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.73 + 4.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.19iT - 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 + (7.50 + 4.33i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.54 + 4.40i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.44 + 3.14i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.45 - 1.99i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 3.07iT - 73T^{2} \) |
| 79 | \( 1 - 2.16T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 + (-11.6 + 6.71i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 6.20i)T + (48.5 + 84.0i)T^{2} \) |
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
−11.56623911106981863533592766176, −10.61808428555820962979939002806, −9.437791672586386536196760080442, −8.786228502069953996608007535462, −8.082339231401628343844657282206, −6.83242242470184136785394177372, −5.17792063285645377141170718148, −4.50796504843666551099939683577, −3.54356239923650835747687544350, −2.28121427738528981144887937509,
1.13947383246798656290786338573, 2.56257017321510831456411048759, 4.44053758292793143765390761317, 5.03048844481583773922989366348, 6.46622547897884141921314191435, 7.27734103881211215993977880238, 8.119387848782755104380868538998, 9.168235394657934706423308478138, 10.23461163461336472210269601820, 10.95543233399909892765564813915