L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (2.63 + 0.189i)7-s − 0.999i·8-s + (−4.67 + 2.69i)11-s − 2.51i·13-s + (−2.19 − 1.48i)14-s + (−0.5 + 0.866i)16-s + (−2.24 − 3.89i)17-s + (−2.48 − 1.43i)19-s + 5.39·22-s + (−0.232 − 0.133i)23-s + (−1.25 + 2.18i)26-s + (1.15 + 2.38i)28-s + 8.89i·29-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.997 + 0.0716i)7-s − 0.353i·8-s + (−1.40 + 0.813i)11-s − 0.698i·13-s + (−0.585 − 0.396i)14-s + (−0.125 + 0.216i)16-s + (−0.545 − 0.945i)17-s + (−0.568 − 0.328i)19-s + 1.15·22-s + (−0.0483 − 0.0279i)23-s + (−0.246 + 0.427i)26-s + (0.218 + 0.449i)28-s + 1.65i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7909116008\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7909116008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.63 - 0.189i)T \) |
good | 11 | \( 1 + (4.67 - 2.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.51iT - 13T^{2} \) |
| 17 | \( 1 + (2.24 + 3.89i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.48 + 1.43i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.232 + 0.133i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 + (-4.18 + 2.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.25 - 5.64i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.760T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 + (3.99 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.27 + 4.19i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.33 - 10.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.27 + 1.31i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.91 - 8.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (10.0 - 5.82i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.29 - 7.44i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.45T + 83T^{2} \) |
| 89 | \( 1 + (3.98 - 6.90i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.16iT - 97T^{2} \) |
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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
−8.599733382363181860233269203542, −8.353709885382156559389614093613, −7.35689747364251707142645449645, −7.04701007064568274190110724719, −5.67197605585838601272716864827, −4.96300234014806645539184851180, −4.30026062852617539503413185818, −2.85392276138053913478938824054, −2.35992896738636617296814834189, −1.13953809679200660032134813389,
0.31615537077247545733565601265, 1.77156187178959820831441426959, 2.50865997447267313464070974330, 3.90864775175476760588257050990, 4.72804545747097940457104925476, 5.62358088289317983078406380346, 6.21604104321865591373854650844, 7.18384607668798252804751819784, 7.993330919766399777393910222762, 8.353110759853428150620417905844