| L(s) = 1 | + (−0.321 + 1.37i)2-s + (−0.342 + 0.939i)3-s + (−1.79 − 0.885i)4-s + (1.48 − 0.262i)5-s + (−1.18 − 0.773i)6-s + (1.07 − 1.86i)7-s + (1.79 − 2.18i)8-s + (−0.766 − 0.642i)9-s + (−0.117 + 2.13i)10-s + (2.84 − 1.64i)11-s + (1.44 − 1.38i)12-s + (1.41 + 3.87i)13-s + (2.22 + 2.08i)14-s + (−0.262 + 1.48i)15-s + (2.43 + 3.17i)16-s + (1.31 − 1.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.227 + 0.973i)2-s + (−0.197 + 0.542i)3-s + (−0.896 − 0.442i)4-s + (0.665 − 0.117i)5-s + (−0.483 − 0.315i)6-s + (0.407 − 0.705i)7-s + (0.634 − 0.772i)8-s + (−0.255 − 0.214i)9-s + (−0.0370 + 0.675i)10-s + (0.858 − 0.495i)11-s + (0.417 − 0.399i)12-s + (0.391 + 1.07i)13-s + (0.594 + 0.557i)14-s + (−0.0677 + 0.384i)15-s + (0.607 + 0.793i)16-s + (0.319 − 0.268i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09067 + 0.759925i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09067 + 0.759925i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.321 - 1.37i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (-4.29 + 0.758i)T \) |
| good | 5 | \( 1 + (-1.48 + 0.262i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.07 + 1.86i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.84 + 1.64i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.41 - 3.87i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.31 + 1.10i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.464 + 2.63i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.48 - 1.76i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.588 - 1.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.85iT - 37T^{2} \) |
| 41 | \( 1 + (3.69 + 1.34i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-9.14 + 1.61i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.24 + 3.55i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.04 - 0.184i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.04 - 8.39i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (4.91 + 0.866i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.902 - 1.07i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.42 + 13.7i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-9.10 - 3.31i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (5.05 + 1.83i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.61 + 1.51i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.20 - 1.16i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-9.85 + 8.27i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.06013060888872918762387700219, −10.09700352925149213867655565814, −9.321530849057235635703483564288, −8.697999807129670779783397011695, −7.46064378782364346972185844059, −6.56127955018044850963964051183, −5.67231508850609879823736881293, −4.66688506375633810480532258152, −3.70951687568219291456731617807, −1.26921530001841893297355531611,
1.30875129247528242201091855328, 2.39834533191247882023797359739, 3.71458179331485936261401838645, 5.24800728190615654147840246422, 5.95692179163884373016920671688, 7.45592490528015843111592462194, 8.297667179211355966627056169292, 9.330657726533587409647440824467, 9.969870963336909436788360197051, 11.04453876256086178595586957036
