L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.231 + 1.31i)3-s + (0.766 − 0.642i)4-s + (−0.231 − 1.31i)6-s + (1.18 + 2.05i)7-s + (−0.500 + 0.866i)8-s + (1.15 + 0.419i)9-s + (−1.33 + 2.31i)11-s + (0.665 + 1.15i)12-s + (−0.168 − 0.955i)13-s + (−1.81 − 1.52i)14-s + (0.173 − 0.984i)16-s + (−5.30 + 1.93i)17-s − 1.22·18-s + (2.94 + 3.21i)19-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.133 + 0.757i)3-s + (0.383 − 0.321i)4-s + (−0.0944 − 0.535i)6-s + (0.448 + 0.776i)7-s + (−0.176 + 0.306i)8-s + (0.384 + 0.139i)9-s + (−0.403 + 0.698i)11-s + (0.192 + 0.332i)12-s + (−0.0467 − 0.265i)13-s + (−0.485 − 0.407i)14-s + (0.0434 − 0.246i)16-s + (−1.28 + 0.468i)17-s − 0.289·18-s + (0.674 + 0.737i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.297373 + 0.934177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.297373 + 0.934177i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.94 - 3.21i)T \) |
good | 3 | \( 1 + (0.231 - 1.31i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-1.18 - 2.05i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.33 - 2.31i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.168 + 0.955i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (5.30 - 1.93i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-6.55 + 5.49i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.0701 + 0.0255i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.00986 - 0.0170i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + (1.90 - 10.7i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.40 - 3.69i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (4.94 + 1.79i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.0681 + 0.0572i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (10.0 - 3.67i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (6.12 - 5.14i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.79 - 1.74i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (7.95 + 6.67i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.47 - 8.38i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.75 + 15.6i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.93 + 5.08i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.149 + 0.847i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (10.4 - 3.79i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.34774549501980768976484151713, −9.502736669472624495953762399825, −8.828105641814124269996381097896, −7.993148017783895562025177863008, −7.10920405570423660096881992449, −6.11349885098779043893560241105, −5.03822518341956549506099031942, −4.46189869928005240672917049979, −2.88188397447336104509593094983, −1.66853458163194531630152429175,
0.59652832429553479152712832944, 1.70471638184581202088501339470, 3.01277220310534198060585413953, 4.27327377563666994686352975737, 5.40725697020189258348544443418, 6.75594199501498169934780045611, 7.17280110685435295496615485282, 7.904867413799898167337402843004, 8.958420345242022865309861112202, 9.559027872654640849597594823208