| L(s) = 1 | + (−0.808 + 0.359i)2-s + (0.558 + 1.63i)3-s + (−0.814 + 0.904i)4-s + (1.09 + 1.94i)5-s + (−1.04 − 1.12i)6-s + (−0.458 + 0.793i)7-s + (0.879 − 2.70i)8-s + (−2.37 + 1.83i)9-s + (−1.58 − 1.18i)10-s + (1.33 − 0.594i)11-s + (−1.93 − 0.830i)12-s + (−1.67 − 0.744i)13-s + (0.0846 − 0.805i)14-s + (−2.58 + 2.88i)15-s + (0.00858 + 0.0816i)16-s + (1.50 − 4.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.571 + 0.254i)2-s + (0.322 + 0.946i)3-s + (−0.407 + 0.452i)4-s + (0.490 + 0.871i)5-s + (−0.425 − 0.458i)6-s + (−0.173 + 0.299i)7-s + (0.310 − 0.956i)8-s + (−0.791 + 0.610i)9-s + (−0.501 − 0.373i)10-s + (0.402 − 0.179i)11-s + (−0.559 − 0.239i)12-s + (−0.464 − 0.206i)13-s + (0.0226 − 0.215i)14-s + (−0.666 + 0.745i)15-s + (0.00214 + 0.0204i)16-s + (0.364 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.264127 + 0.841158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.264127 + 0.841158i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.558 - 1.63i)T \) |
| 5 | \( 1 + (-1.09 - 1.94i)T \) |
| good | 2 | \( 1 + (0.808 - 0.359i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (0.458 - 0.793i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.33 + 0.594i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (1.67 + 0.744i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.50 + 4.62i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.04 - 6.28i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.0683 - 0.650i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-6.29 - 1.33i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-2.87 + 0.611i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (4.86 - 3.53i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.26 - 1.00i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-2.01 + 3.48i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.26 - 1.11i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (3.46 + 10.6i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.82 - 0.813i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-11.5 + 5.14i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-8.34 + 1.77i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (-5.09 - 15.6i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.01 + 1.46i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (6.05 + 1.28i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.67 - 2.97i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (11.4 + 8.29i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.89 + 1.25i)T + (88.6 + 39.4i)T^{2} \) |
| 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
−12.59265678153859517745144246312, −11.50785421525326168263249046187, −10.16282909243825889542213212063, −9.850732504923217165973383420453, −8.818467031554305635186372854159, −7.896482449767372852250444311209, −6.70175415561252805437002844980, −5.34534928450508689010760860927, −3.89654840065909717759322181568, −2.80742854982048963437674780918,
0.905272883443158279539269227828, 2.23641285588597575455536907684, 4.42583269424013921770842317331, 5.72210150333086207140808594856, 6.84244888406560935449442185234, 8.220500023932639031896591982785, 8.877900484439824731617437486161, 9.702595799455384215452628902458, 10.74691908556652286120258024686, 12.02456486585056767285585428092
