L(s) = 1 | + (0.576 + 1.29i)2-s + (−1.33 + 1.48i)4-s + (1.88 + 3.26i)5-s + (−2.23 − 1.41i)7-s + (−2.69 − 0.868i)8-s + (−3.12 + 4.30i)10-s + (1.47 + 0.849i)11-s + 5.64i·13-s + (0.543 − 3.70i)14-s + (−0.428 − 3.97i)16-s + (−2.26 − 1.30i)17-s + (−1.18 − 2.04i)19-s + (−7.36 − 1.55i)20-s + (−0.249 + 2.38i)22-s + (−0.653 − 1.13i)23-s + ⋯ |
L(s) = 1 | + (0.407 + 0.913i)2-s + (−0.668 + 0.743i)4-s + (0.841 + 1.45i)5-s + (−0.844 − 0.535i)7-s + (−0.951 − 0.307i)8-s + (−0.988 + 1.36i)10-s + (0.443 + 0.256i)11-s + 1.56i·13-s + (0.145 − 0.989i)14-s + (−0.107 − 0.994i)16-s + (−0.548 − 0.316i)17-s + (−0.270 − 0.469i)19-s + (−1.64 − 0.347i)20-s + (−0.0532 + 0.509i)22-s + (−0.136 − 0.236i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.248045 + 1.51120i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.248045 + 1.51120i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.576 - 1.29i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.23 + 1.41i)T \) |
good | 5 | \( 1 + (-1.88 - 3.26i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.47 - 0.849i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.64iT - 13T^{2} \) |
| 17 | \( 1 + (2.26 + 1.30i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.18 + 2.04i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.653 + 1.13i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + (-4.75 - 2.74i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.20 - 3.00i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.10iT - 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 + (-3.53 - 6.12i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.488 - 0.846i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.67 - 3.85i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.64 + 4.41i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.69 + 11.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.40T + 71T^{2} \) |
| 73 | \( 1 + (1.09 - 1.88i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (15.0 - 8.71i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.8iT - 83T^{2} \) |
| 89 | \( 1 + (-9.58 + 5.53i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−11.32011402616151775315719496840, −10.21848096214408525307775818531, −9.558364432917784367357842824125, −8.650278108932873283241303545674, −7.05153906188573320805336343994, −6.76350601476759255019276783669, −6.22544791398189700414146263281, −4.71089208533452627385912714984, −3.62156362139712202947003995249, −2.47684141423287223325914830152,
0.815685338238425173882795941596, 2.25739738266524139200017921750, 3.53779807563559044694105838533, 4.82285373731343702724640404664, 5.64606379928139226163046191078, 6.31399473388798174607560659883, 8.430262217338610141092503338161, 8.785128652605703601386332334815, 9.954390268508284372358451672657, 10.18224393340670642714858165298