| L(s) = 1 | + (0.438 − 2.67i)2-s + (1.10 − 1.33i)3-s + (−5.07 − 1.71i)4-s + (−2.43 + 1.64i)5-s + (−3.09 − 3.53i)6-s + (3.44 − 0.757i)7-s + (−4.26 + 8.05i)8-s + (−0.569 − 2.94i)9-s + (3.34 + 7.22i)10-s + (1.00 − 0.766i)11-s + (−7.88 + 4.89i)12-s + (−1.15 − 1.92i)13-s + (−0.517 − 9.54i)14-s + (−0.477 + 5.06i)15-s + (11.1 + 8.47i)16-s + (0.288 − 1.31i)17-s + ⋯ |
| L(s) = 1 | + (0.310 − 1.89i)2-s + (0.636 − 0.771i)3-s + (−2.53 − 0.855i)4-s + (−1.08 + 0.736i)5-s + (−1.26 − 1.44i)6-s + (1.30 − 0.286i)7-s + (−1.50 + 2.84i)8-s + (−0.189 − 0.981i)9-s + (1.05 + 2.28i)10-s + (0.304 − 0.231i)11-s + (−2.27 + 1.41i)12-s + (−0.321 − 0.534i)13-s + (−0.138 − 2.55i)14-s + (−0.123 + 1.30i)15-s + (2.78 + 2.11i)16-s + (0.0700 − 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0741249 + 1.28523i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0741249 + 1.28523i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.10 + 1.33i)T \) |
| 59 | \( 1 + (-7.18 + 2.70i)T \) |
| good | 2 | \( 1 + (-0.438 + 2.67i)T + (-1.89 - 0.638i)T^{2} \) |
| 5 | \( 1 + (2.43 - 1.64i)T + (1.85 - 4.64i)T^{2} \) |
| 7 | \( 1 + (-3.44 + 0.757i)T + (6.35 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-1.00 + 0.766i)T + (2.94 - 10.5i)T^{2} \) |
| 13 | \( 1 + (1.15 + 1.92i)T + (-6.08 + 11.4i)T^{2} \) |
| 17 | \( 1 + (-0.288 + 1.31i)T + (-15.4 - 7.13i)T^{2} \) |
| 19 | \( 1 + (-5.94 + 0.646i)T + (18.5 - 4.08i)T^{2} \) |
| 23 | \( 1 + (-1.63 - 1.92i)T + (-3.72 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.67 - 0.274i)T + (27.4 - 9.25i)T^{2} \) |
| 31 | \( 1 + (0.831 - 7.64i)T + (-30.2 - 6.66i)T^{2} \) |
| 37 | \( 1 + (1.69 - 0.900i)T + (20.7 - 30.6i)T^{2} \) |
| 41 | \( 1 + (-4.74 - 4.03i)T + (6.63 + 40.4i)T^{2} \) |
| 43 | \( 1 + (0.654 - 0.860i)T + (-11.5 - 41.4i)T^{2} \) |
| 47 | \( 1 + (-2.49 + 3.67i)T + (-17.3 - 43.6i)T^{2} \) |
| 53 | \( 1 + (1.79 - 3.87i)T + (-34.3 - 40.3i)T^{2} \) |
| 61 | \( 1 + (9.31 + 1.52i)T + (57.8 + 19.4i)T^{2} \) |
| 67 | \( 1 + (-5.34 - 2.83i)T + (37.5 + 55.4i)T^{2} \) |
| 71 | \( 1 + (-1.23 - 0.839i)T + (26.2 + 65.9i)T^{2} \) |
| 73 | \( 1 + (-3.46 + 0.187i)T + (72.5 - 7.89i)T^{2} \) |
| 79 | \( 1 + (-2.00 - 7.20i)T + (-67.6 + 40.7i)T^{2} \) |
| 83 | \( 1 + (12.3 + 11.6i)T + (4.49 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-0.561 - 3.42i)T + (-84.3 + 28.4i)T^{2} \) |
| 97 | \( 1 + (9.37 + 0.508i)T + (96.4 + 10.4i)T^{2} \) |
| 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
−11.85717513929202163154111717173, −11.50900446067886946898217518433, −10.62881986800071274611429457395, −9.334809380628658385853161616589, −8.220163570615496691300121493083, −7.36438235077393380059753197931, −5.12161496968438384013538368784, −3.74131006166365348276390580563, −2.87063638798918398234441311645, −1.23601213192556079795273389261,
3.94459616680333517754974241054, 4.64049375047722479914151814930, 5.49265429370846617136201679995, 7.38627524861102223307527039361, 7.976474646601661822981025692345, 8.734470759285416742156105542931, 9.539389342253620152114945924857, 11.41724396836553987678367220989, 12.48550128238594551143003432686, 13.77040554311289950099856646115
