L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.207 − 0.358i)5-s + (−2.62 − 0.358i)7-s + 0.999·8-s + (0.207 + 0.358i)10-s + (−0.5 − 0.866i)11-s − 1.17·13-s + (1.62 − 2.09i)14-s + (−0.5 + 0.866i)16-s + (1.08 + 1.88i)17-s + (−0.414 + 0.717i)19-s − 0.414·20-s + 0.999·22-s + (1.62 − 2.80i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0926 − 0.160i)5-s + (−0.990 − 0.135i)7-s + 0.353·8-s + (0.0654 + 0.113i)10-s + (−0.150 − 0.261i)11-s − 0.324·13-s + (0.433 − 0.558i)14-s + (−0.125 + 0.216i)16-s + (0.263 + 0.456i)17-s + (−0.0950 + 0.164i)19-s − 0.0926·20-s + 0.213·22-s + (0.338 − 0.585i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9902969648\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9902969648\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.62 + 0.358i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.207 + 0.358i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 + (-1.08 - 1.88i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.414 - 0.717i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.62 + 2.80i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + (-3.24 - 5.61i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.82 - 8.36i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + (-4.62 + 8.00i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.58 - 4.47i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.82 - 3.16i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.792 + 1.37i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.74 - 11.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + (-2.41 - 4.18i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.37 - 4.11i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.82T + 83T^{2} \) |
| 89 | \( 1 + (6.24 - 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−9.819700189522624510861094223997, −8.774630435469290771339669069384, −8.340372455464620669009341998868, −7.13298533054877379104933178456, −6.68078713577740265998364539213, −5.73394796241594168143685231731, −4.92718593687242269197684456522, −3.75773428502498641542591272061, −2.70190915678549223913815716721, −1.04621127453767368454729911395,
0.55407289796744983334991458105, 2.24354150718876193339065855400, 3.02829206909068152429715690932, 4.04172649309179217644802879903, 5.11056901136977038405331426200, 6.16901182453900406748280458673, 7.02897476909572800431686160951, 7.80542594501443287958165685185, 8.825608951632944417830845977678, 9.529818951296537360587761561041