L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.17 + 1.27i)3-s + (−0.499 − 0.866i)4-s + (0.870 − 0.502i)5-s + (−0.517 − 1.65i)6-s + (1.33 + 2.28i)7-s + 0.999·8-s + (−0.249 − 2.98i)9-s + 1.00i·10-s + 0.620·11-s + (1.69 + 0.378i)12-s + (1.14 + 3.41i)13-s + (−2.64 + 0.0151i)14-s + (−0.380 + 1.69i)15-s + (−0.5 + 0.866i)16-s + (0.171 + 0.296i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.677 + 0.735i)3-s + (−0.249 − 0.433i)4-s + (0.389 − 0.224i)5-s + (−0.211 − 0.674i)6-s + (0.504 + 0.863i)7-s + 0.353·8-s + (−0.0831 − 0.996i)9-s + 0.317i·10-s + 0.187·11-s + (0.487 + 0.109i)12-s + (0.316 + 0.948i)13-s + (−0.707 + 0.00404i)14-s + (−0.0981 + 0.438i)15-s + (−0.125 + 0.216i)16-s + (0.0415 + 0.0720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.394460 + 0.900886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.394460 + 0.900886i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (1.17 - 1.27i)T \) |
| 7 | \( 1 + (-1.33 - 2.28i)T \) |
| 13 | \( 1 + (-1.14 - 3.41i)T \) |
good | 5 | \( 1 + (-0.870 + 0.502i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 0.620T + 11T^{2} \) |
| 17 | \( 1 + (-0.171 - 0.296i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - 4.33T + 19T^{2} \) |
| 23 | \( 1 + (-2.44 - 1.41i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (8.23 - 4.75i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.25 + 2.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.76 + 2.17i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.47 - 4.31i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.602 - 1.04i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.0442 + 0.0255i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.15 - 2.39i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.67 + 1.54i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 6.68iT - 61T^{2} \) |
| 67 | \( 1 - 5.48iT - 67T^{2} \) |
| 71 | \( 1 + (0.621 - 1.07i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.46 + 7.72i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.458 - 0.793i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.2iT - 83T^{2} \) |
| 89 | \( 1 + (-3.11 - 1.80i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.10 + 8.84i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.25322782048277876387430267520, −10.01841543730764584772906320568, −9.229410360778372635217889846036, −8.825933801007747537437034209410, −7.47760434617031168666265714886, −6.42045756202627678933379274896, −5.50121040876197760466944573354, −4.97048698749565753808621089785, −3.61440100076441367259617170192, −1.59988617314825328028562975807,
0.75446882387470049111612512599, 1.99909602450886702715694302829, 3.47770790341713073370323522005, 4.88023457152607828214620071662, 5.85713562497641217780839014104, 7.04383598255988945035376804846, 7.71419581688254179277770962336, 8.628371098508247969719524322891, 10.00933851735795309049220527935, 10.48431175754542947111698788072