| L(s) = 1 | + (1.53 − 2.16i)3-s + (−1.11 + 0.214i)5-s + (−0.183 − 2.63i)7-s + (−1.32 − 3.82i)9-s + (1.39 − 1.09i)11-s + (−0.0127 + 0.00821i)13-s + (−1.25 + 2.74i)15-s + (−1.42 − 1.35i)17-s + (−1.89 + 1.80i)19-s + (−5.98 − 3.66i)21-s + (0.937 − 4.70i)23-s + (−3.44 + 1.37i)25-s + (−2.66 − 0.782i)27-s + (0.499 − 0.146i)29-s + (2.62 − 0.250i)31-s + ⋯ |
| L(s) = 1 | + (0.888 − 1.24i)3-s + (−0.498 + 0.0961i)5-s + (−0.0692 − 0.997i)7-s + (−0.441 − 1.27i)9-s + (0.419 − 0.330i)11-s + (−0.00354 + 0.00227i)13-s + (−0.323 + 0.707i)15-s + (−0.344 − 0.328i)17-s + (−0.434 + 0.413i)19-s + (−1.30 − 0.800i)21-s + (0.195 − 0.980i)23-s + (−0.688 + 0.275i)25-s + (−0.512 − 0.150i)27-s + (0.0927 − 0.0272i)29-s + (0.471 − 0.0450i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.780585 - 1.47694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.780585 - 1.47694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.183 + 2.63i)T \) |
| 23 | \( 1 + (-0.937 + 4.70i)T \) |
| good | 3 | \( 1 + (-1.53 + 2.16i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (1.11 - 0.214i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-1.39 + 1.09i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (0.0127 - 0.00821i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.42 + 1.35i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (1.89 - 1.80i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-0.499 + 0.146i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.62 + 0.250i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (3.24 + 9.36i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-3.97 - 4.58i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.18 - 9.16i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-3.03 + 5.26i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.206 + 4.33i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-7.17 - 3.69i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-1.25 - 1.75i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (4.46 - 1.78i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.873 - 6.07i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.19 - 4.93i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.389 + 8.17i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-1.91 + 2.20i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-17.3 - 1.66i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (0.0785 + 0.0906i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.29068863380383066163259417702, −9.141378509051160566047140954112, −8.325814870396137075408933890281, −7.59411230719591260478122774312, −6.96869772592094196463176678558, −6.12018869419778791995513129946, −4.39397027692811446796927887039, −3.45626031281654957291173066522, −2.24228931006497188576609883526, −0.833929951085611900139397669500,
2.22997897072940148308106981119, 3.36331618057345870468506966965, 4.22701619053621111463671711311, 5.11525774833472174688900425985, 6.31007282795169198103850852293, 7.62686212761510629604021088172, 8.593068662354757313123499474812, 9.063199340363533976017227046411, 9.828177484456950025689196374681, 10.68755092767003297563979423450
