L(s) = 1 | + (1.90 − 1.38i)2-s + (0.885 + 2.72i)3-s + (1.08 − 3.34i)4-s + (0.809 + 0.587i)5-s + (5.44 + 3.95i)6-s + (1.04 − 3.21i)7-s + (−1.10 − 3.38i)8-s + (−4.21 + 3.06i)9-s + 2.34·10-s + 10.0·12-s + (1.19 − 0.871i)13-s + (−2.45 − 7.55i)14-s + (−0.885 + 2.72i)15-s + (−1.08 − 0.785i)16-s + (−3.01 − 2.19i)17-s + (−3.78 + 11.6i)18-s + ⋯ |
L(s) = 1 | + (1.34 − 0.976i)2-s + (0.511 + 1.57i)3-s + (0.543 − 1.67i)4-s + (0.361 + 0.262i)5-s + (2.22 + 1.61i)6-s + (0.395 − 1.21i)7-s + (−0.389 − 1.19i)8-s + (−1.40 + 1.02i)9-s + 0.742·10-s + 2.91·12-s + (0.332 − 0.241i)13-s + (−0.656 − 2.02i)14-s + (−0.228 + 0.703i)15-s + (−0.270 − 0.196i)16-s + (−0.732 − 0.532i)17-s + (−0.892 + 2.74i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.66656 - 0.450717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.66656 - 0.450717i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.90 + 1.38i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.885 - 2.72i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-1.04 + 3.21i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-1.19 + 0.871i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.01 + 2.19i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.993 - 3.05i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 + (1.45 - 4.46i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (7.45 - 5.41i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.833 - 2.56i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.40 + 4.32i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 + (3.25 + 10.0i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.03 + 1.47i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.68 + 11.3i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (11.0 + 8.04i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 7.83T + 67T^{2} \) |
| 71 | \( 1 + (2.03 + 1.47i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.916 + 2.82i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.47 + 3.97i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.10 - 4.43i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + (8.91 - 6.47i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.70469733430854149999008068766, −10.19322981533803876041014541918, −9.326281943482762156865131420878, −8.169372412127491527390953412731, −6.75952880485145628938966355121, −5.33915750769978216646516437116, −4.80927945704964615903342630503, −3.70284697376768136855742251645, −3.43133531859961116046659725146, −1.91993562946127223771998850631,
1.86247716612169825439855667381, 2.80660902112801163439018899173, 4.28727914529893494938254422846, 5.57659540481148318457948064586, 6.08323778334427243817992372727, 6.90990568010312411810250783223, 7.80542392948154740840426549211, 8.496540126973753131882259530497, 9.333605138175833047827543380809, 11.27764834636393770276424509221