| L(s) = 1 | + (−0.682 + 1.73i)2-s + (0.970 + 1.43i)3-s + (−1.09 − 1.01i)4-s + (−0.565 + 0.272i)5-s + (−3.15 + 0.709i)6-s + (−2.38 − 1.14i)7-s + (−0.852 + 0.410i)8-s + (−1.11 + 2.78i)9-s + (−0.0876 − 1.16i)10-s + (−1.43 − 1.79i)11-s + (0.394 − 2.55i)12-s + (−0.0553 + 0.141i)13-s + (3.62 − 3.36i)14-s + (−0.939 − 0.546i)15-s + (−0.355 − 4.74i)16-s + (−4.56 + 4.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.482 + 1.23i)2-s + (0.560 + 0.828i)3-s + (−0.547 − 0.507i)4-s + (−0.252 + 0.121i)5-s + (−1.28 + 0.289i)6-s + (−0.900 − 0.434i)7-s + (−0.301 + 0.145i)8-s + (−0.371 + 0.928i)9-s + (−0.0277 − 0.369i)10-s + (−0.432 − 0.542i)11-s + (0.113 − 0.738i)12-s + (−0.0153 + 0.0391i)13-s + (0.969 − 0.898i)14-s + (−0.242 − 0.141i)15-s + (−0.0888 − 1.18i)16-s + (−1.10 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.312905 - 0.529114i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.312905 - 0.529114i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.970 - 1.43i)T \) |
| 7 | \( 1 + (2.38 + 1.14i)T \) |
| good | 2 | \( 1 + (0.682 - 1.73i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (0.565 - 0.272i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (1.43 + 1.79i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.0553 - 0.141i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (4.56 - 4.23i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.98 + 3.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.14 - 5.01i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-4.55 - 1.40i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (2.14 - 3.72i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.69 + 0.830i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-7.50 - 5.11i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-6.56 + 4.47i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (1.56 - 3.98i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (7.82 - 2.41i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-8.62 + 5.87i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-2.24 + 2.08i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.672 - 1.16i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.851 - 3.72i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (8.10 + 1.22i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-6.42 - 11.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.57 + 6.56i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-3.57 - 9.10i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-6.19 + 10.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.38384795311808741568901836742, −10.61747048690761298156881143363, −9.542660121487233482519705608540, −8.969203840776048608425773459643, −8.054009205342079521238762559666, −7.26268863515397508277929591863, −6.26944571206591098356288381553, −5.25636073416508498871685772063, −3.89895553537567407474811131477, −2.85522952709913561263385082007,
0.39374933995012399801073007913, 2.18633082096710951841846123172, 2.84607591095744466400887120525, 4.12821031406497146355182135897, 5.97423410648297474771775053203, 6.88549626731204798382639017820, 8.019655860328040991226305476312, 8.962264160995844113288499429195, 9.606376983257215928790758447831, 10.44097329080728124920693086360
