L(s) = 1 | + (2.44 + 1.57i)3-s + (−0.415 + 0.909i)5-s + (2.39 + 0.702i)7-s + (2.26 + 4.95i)9-s + (−1.36 + 1.57i)11-s + (−4.61 + 1.35i)13-s + (−2.44 + 1.57i)15-s + (−0.933 − 6.49i)17-s + (0.880 − 6.12i)19-s + (4.74 + 5.47i)21-s + (2.85 + 3.85i)23-s + (−0.654 − 0.755i)25-s + (−1.01 + 7.04i)27-s + (0.00773 + 0.0537i)29-s + (9.13 − 5.87i)31-s + ⋯ |
L(s) = 1 | + (1.41 + 0.907i)3-s + (−0.185 + 0.406i)5-s + (0.904 + 0.265i)7-s + (0.754 + 1.65i)9-s + (−0.412 + 0.475i)11-s + (−1.28 + 0.376i)13-s + (−0.631 + 0.405i)15-s + (−0.226 − 1.57i)17-s + (0.201 − 1.40i)19-s + (1.03 + 1.19i)21-s + (0.595 + 0.803i)23-s + (−0.130 − 0.151i)25-s + (−0.194 + 1.35i)27-s + (0.00143 + 0.00998i)29-s + (1.64 − 1.05i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.79299 + 1.25469i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79299 + 1.25469i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-2.85 - 3.85i)T \) |
good | 3 | \( 1 + (-2.44 - 1.57i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (-2.39 - 0.702i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (1.36 - 1.57i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.61 - 1.35i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.933 + 6.49i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.880 + 6.12i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.00773 - 0.0537i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-9.13 + 5.87i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.12 - 6.83i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (1.35 - 2.97i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (4.58 + 2.94i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + (-4.64 - 1.36i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (13.3 - 3.91i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-8.36 + 5.37i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.47 - 5.16i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-3.07 - 3.54i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.727 + 5.06i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-0.767 + 0.225i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (3.09 + 6.76i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (3.96 + 2.54i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.12 + 6.84i)T + (-63.5 - 73.3i)T^{2} \) |
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
−11.26718200350457909311242003461, −9.906748332819360505765969283666, −9.590622745564568544716453652483, −8.594269571021336380556206480213, −7.70484503786115747652634523770, −7.02104111548780464237563349882, −4.87008091753196613270341598502, −4.66523811769618704970123059099, −2.98592486803711836537543206202, −2.37287470455075272263396576484,
1.38448007456420114738403375694, 2.54277480338818249938427130645, 3.78590592464195129542474053562, 5.02247223963658862342108063312, 6.45419726747159054259722643163, 7.64217219210778317573550853040, 8.133097865292779973905067692018, 8.645419882224046046248690471135, 9.889986134431624556423779633436, 10.78504383479792226941854889461