L(s) = 1 | + (0.942 − 2.27i)3-s + (−2.49 + 1.03i)5-s + (−1.37 + 1.37i)7-s + (−2.17 − 2.17i)9-s + (1.70 + 4.12i)11-s + (4.86 + 2.01i)13-s + 6.66i·15-s − 6.31i·17-s + (4.72 + 1.95i)19-s + (1.82 + 4.41i)21-s + (0.288 + 0.288i)23-s + (1.63 − 1.63i)25-s + (−0.159 + 0.0660i)27-s + (0.428 − 1.03i)29-s − 3.69·31-s + ⋯ |
L(s) = 1 | + (0.544 − 1.31i)3-s + (−1.11 + 0.462i)5-s + (−0.518 + 0.518i)7-s + (−0.723 − 0.723i)9-s + (0.515 + 1.24i)11-s + (1.34 + 0.558i)13-s + 1.72i·15-s − 1.53i·17-s + (1.08 + 0.449i)19-s + (0.398 + 0.963i)21-s + (0.0602 + 0.0602i)23-s + (0.327 − 0.327i)25-s + (−0.0306 + 0.0127i)27-s + (0.0796 − 0.192i)29-s − 0.663·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.592020976\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.592020976\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.942 + 2.27i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (2.49 - 1.03i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (1.37 - 1.37i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.70 - 4.12i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-4.86 - 2.01i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 6.31iT - 17T^{2} \) |
| 19 | \( 1 + (-4.72 - 1.95i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.288 - 0.288i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.428 + 1.03i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 3.69T + 31T^{2} \) |
| 37 | \( 1 + (-10.4 + 4.32i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-1.68 - 1.68i)T + 41iT^{2} \) |
| 43 | \( 1 + (-1.67 - 4.03i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 4.83iT - 47T^{2} \) |
| 53 | \( 1 + (-0.207 - 0.501i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (7.25 - 3.00i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (1.40 - 3.39i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (0.123 - 0.299i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-6.54 + 6.54i)T - 71iT^{2} \) |
| 73 | \( 1 + (-5.53 - 5.53i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.877iT - 79T^{2} \) |
| 83 | \( 1 + (-3.42 - 1.41i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.46 + 6.46i)T - 89iT^{2} \) |
| 97 | \( 1 + 3.58T + 97T^{2} \) |
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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
−9.537144524884837689870536213982, −9.131490935165693189389126521314, −7.890936234707903079633035094437, −7.49277443812576561912678900755, −6.80187752364150353556562243829, −5.98780368661630086158122196971, −4.45971021500813938195761887533, −3.41033333583343604196060017521, −2.48791210522614570283918441271, −1.17173877854830280520946771290,
0.865459075323616257462728658211, 3.33326928898893148182274231496, 3.61687354051567971991701238965, 4.31817266570394639069395681839, 5.54784080237797875557372109271, 6.49505877722952503178893085208, 7.86670149278277095857145690124, 8.446602538863382360765443382738, 9.043344583951703669476506308103, 9.929640444123692661940489879603