L(s) = 1 | + (0.0713 − 0.997i)2-s + (−0.977 + 0.212i)3-s + (−0.989 − 0.142i)4-s + (1.21 − 1.87i)5-s + (0.142 + 0.989i)6-s + (3.65 − 1.99i)7-s + (−0.212 + 0.977i)8-s + (0.909 − 0.415i)9-s + (−1.78 − 1.34i)10-s + (1.81 + 1.57i)11-s + (0.997 − 0.0713i)12-s + (−1.16 + 2.13i)13-s + (−1.72 − 3.78i)14-s + (−0.791 + 2.09i)15-s + (0.959 + 0.281i)16-s + (1.68 + 1.26i)17-s + ⋯ |
L(s) = 1 | + (0.0504 − 0.705i)2-s + (−0.564 + 0.122i)3-s + (−0.494 − 0.0711i)4-s + (0.544 − 0.838i)5-s + (0.0580 + 0.404i)6-s + (1.38 − 0.754i)7-s + (−0.0751 + 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.563 − 0.426i)10-s + (0.546 + 0.473i)11-s + (0.287 − 0.0205i)12-s + (−0.322 + 0.590i)13-s + (−0.462 − 1.01i)14-s + (−0.204 + 0.539i)15-s + (0.239 + 0.0704i)16-s + (0.408 + 0.306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05156 - 1.17957i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05156 - 1.17957i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 3 | \( 1 + (0.977 - 0.212i)T \) |
| 5 | \( 1 + (-1.21 + 1.87i)T \) |
| 23 | \( 1 + (-4.76 - 0.560i)T \) |
good | 7 | \( 1 + (-3.65 + 1.99i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-1.81 - 1.57i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.16 - 2.13i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-1.68 - 1.26i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 4.91i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-3.75 + 0.539i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (8.69 - 5.58i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.03 + 0.758i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-3.56 + 7.80i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.190 + 0.874i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (5.40 + 5.40i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.57 - 8.38i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (3.47 + 11.8i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (6.51 + 10.1i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (10.9 + 0.779i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-3.90 - 4.50i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-3.88 - 5.19i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (0.848 - 0.249i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.763 - 2.04i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-10.1 - 6.55i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.95 - 13.2i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.49360171316901385346215066582, −9.371551006732187764187702500703, −8.858615060119239755044583607036, −7.65403687738425994665957519835, −6.69964917907942392119008962109, −5.16428851621272259601915869886, −4.87733775525380465327385793092, −3.87527476561612103704118714060, −1.96109475435205757584661006496, −1.04722514950216705951997233189,
1.55813054764265812333296480153, 3.08510716658844792078707334829, 4.61018181318555316553189729234, 5.60804579273674538683012213350, 6.00752408500318421393096241779, 7.21633483784390179087469793194, 7.88779973993131263660317458013, 8.867907303387674720008834921844, 9.823872456049932666524703172180, 10.78140702874331611075189018266