| L(s) = 1 | + (0.943 + 1.05i)2-s + (0.342 − 0.939i)3-s + (−0.219 + 1.98i)4-s + (3.30 − 0.582i)5-s + (1.31 − 0.526i)6-s + (−1.96 + 3.40i)7-s + (−2.30 + 1.64i)8-s + (−0.766 − 0.642i)9-s + (3.73 + 2.93i)10-s + (−2.32 + 1.34i)11-s + (1.79 + 0.885i)12-s + (1.43 + 3.95i)13-s + (−5.43 + 1.14i)14-s + (0.582 − 3.30i)15-s + (−3.90 − 0.870i)16-s + (4.40 − 3.69i)17-s + ⋯ |
| L(s) = 1 | + (0.667 + 0.744i)2-s + (0.197 − 0.542i)3-s + (−0.109 + 0.993i)4-s + (1.47 − 0.260i)5-s + (0.535 − 0.214i)6-s + (−0.742 + 1.28i)7-s + (−0.813 + 0.581i)8-s + (−0.255 − 0.214i)9-s + (1.18 + 0.926i)10-s + (−0.701 + 0.404i)11-s + (0.517 + 0.255i)12-s + (0.399 + 1.09i)13-s + (−1.45 + 0.304i)14-s + (0.150 − 0.853i)15-s + (−0.976 − 0.217i)16-s + (1.06 − 0.896i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88935 + 1.37362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88935 + 1.37362i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.943 - 1.05i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 19 | \( 1 + (-3.87 + 2.00i)T \) |
| good | 5 | \( 1 + (-3.30 + 0.582i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.96 - 3.40i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.32 - 1.34i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.43 - 3.95i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-4.40 + 3.69i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.958 + 5.43i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.439 + 0.524i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.25 + 2.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.16iT - 37T^{2} \) |
| 41 | \( 1 + (10.3 + 3.77i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (12.0 - 2.11i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.55 - 3.82i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-9.75 - 1.72i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.47 - 6.53i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.92 + 0.340i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.73 + 3.26i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.84 + 10.4i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (13.9 + 5.06i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (3.82 + 1.39i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.58 - 0.912i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.91 - 0.698i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.85 + 7.43i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.80276415035157046407906937218, −10.03228274317659348958938319846, −9.199454877319652837848471919282, −8.646261953043360627349159406439, −7.28536122843458423155891261260, −6.42568352355622241628072157103, −5.67746538558740311780259781101, −4.96544414938860308526765001848, −3.03528081842992944103600669635, −2.18823516229330887311240932469,
1.34713304133028920419604944752, 3.09098401400316717859887724834, 3.58256138755400284453694345851, 5.29239594854168596626613907315, 5.74910502724279016961573580277, 6.91377250258901796443165568109, 8.355241420491502675126969359322, 9.793464872587214133141243198924, 10.22361281417959040188974098496, 10.39294546328608932025422183505
