| L(s) = 1 | + (1.00 + 0.581i)2-s + (−0.322 − 0.559i)4-s − 3.07i·8-s + (5.68 − 3.28i)11-s + (1.14 − 1.98i)16-s + 7.64·22-s + (1.65 + 0.957i)23-s + (2.5 + 4.33i)25-s − 8.89i·29-s + (−3.02 + 1.74i)32-s + (−5.29 + 9.16i)37-s − 5.29·43-s + (−3.67 − 2.12i)44-s + (1.11 + 1.93i)46-s + 5.81i·50-s + ⋯ |
| L(s) = 1 | + (0.712 + 0.411i)2-s + (−0.161 − 0.279i)4-s − 1.08i·8-s + (1.71 − 0.990i)11-s + (0.286 − 0.496i)16-s + 1.63·22-s + (0.345 + 0.199i)23-s + (0.5 + 0.866i)25-s − 1.65i·29-s + (−0.534 + 0.308i)32-s + (−0.869 + 1.50i)37-s − 0.806·43-s + (−0.553 − 0.319i)44-s + (0.164 + 0.284i)46-s + 0.822i·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93549 - 0.360548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93549 - 0.360548i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + (-1.00 - 0.581i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.68 + 3.28i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.65 - 0.957i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.89iT - 29T^{2} \) |
| 31 | \( 1 + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.29 - 9.16i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.357 - 0.206i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15.0iT - 71T^{2} \) |
| 73 | \( 1 + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−11.28622520627307923422464991127, −10.06519116681913159840589954517, −9.273637567784052803948691987634, −8.414283579218747635121702348512, −6.96891052786039680053501119661, −6.30937708296599226529040664042, −5.39944012568569577436602491624, −4.25364166050230831734022363979, −3.36071116789897614056847804500, −1.19292053224765028176085116692,
1.83451747351371218843714023134, 3.31555909477116528978223536273, 4.25015500570082797641519024672, 5.10406532118052524621961895514, 6.48667784933060835322262481708, 7.35816349872892386601431772607, 8.662531130678658322296706272809, 9.242189325831636166529887501488, 10.47504905958964051702794421845, 11.40918048210201114885604459892
