L(s) = 1 | + (−1.72 + 0.128i)3-s + (2.43 + 1.40i)5-s + (−0.717 + 2.54i)7-s + (2.96 − 0.444i)9-s + (2.29 − 1.32i)11-s + (4.59 − 2.65i)13-s + (−4.38 − 2.11i)15-s + (−2.35 − 1.36i)17-s + (0.274 + 0.474i)19-s + (0.911 − 4.49i)21-s + (1.98 + 1.14i)23-s + (1.45 + 2.51i)25-s + (−5.06 + 1.15i)27-s + (−3.72 + 6.45i)29-s + 2.76·31-s + ⋯ |
L(s) = 1 | + (−0.997 + 0.0743i)3-s + (1.08 + 0.628i)5-s + (−0.271 + 0.962i)7-s + (0.988 − 0.148i)9-s + (0.690 − 0.398i)11-s + (1.27 − 0.735i)13-s + (−1.13 − 0.546i)15-s + (−0.571 − 0.329i)17-s + (0.0628 + 0.108i)19-s + (0.198 − 0.980i)21-s + (0.413 + 0.238i)23-s + (0.290 + 0.503i)25-s + (−0.975 + 0.221i)27-s + (−0.692 + 1.19i)29-s + 0.497·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.495429219\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495429219\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 - 0.128i)T \) |
| 7 | \( 1 + (0.717 - 2.54i)T \) |
good | 5 | \( 1 + (-2.43 - 1.40i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.29 + 1.32i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.59 + 2.65i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.35 + 1.36i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.274 - 0.474i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.98 - 1.14i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.72 - 6.45i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + (-1.81 - 3.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.12 + 4.11i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.93 + 2.27i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + (2.53 - 4.38i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 - 14.4iT - 61T^{2} \) |
| 67 | \( 1 - 10.8iT - 67T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (-4.29 - 2.47i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 5.61iT - 79T^{2} \) |
| 83 | \( 1 + (0.719 - 1.24i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.24 + 1.87i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.7 + 9.09i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.24398028513506077827354859679, −9.271286335420661495228166294829, −8.733498726695902107846331205773, −7.26734888936066997474764880374, −6.33470307745286715695354700624, −5.93025337647902263749070452228, −5.24090238993116571010624017272, −3.80949522542363355737583460475, −2.63589485026228676838819875784, −1.27458453920667622395337606988,
0.948205267961119837591635565972, 1.89794560138129249781254047656, 3.93907881303778106107756117699, 4.53829867665234404530704209447, 5.71955645151651253500259427771, 6.38261959925066765125893348623, 6.97451362754615617989300754757, 8.205225912175293744302644936911, 9.468700572892167631167850605428, 9.624921936124591921525230110851