| L(s) = 1 | + (−1.31 − 0.531i)2-s + (−0.947 + 2.60i)3-s + (1.43 + 1.39i)4-s + (−1.56 + 0.275i)5-s + (2.62 − 2.90i)6-s + (−0.102 + 0.176i)7-s + (−1.14 − 2.58i)8-s + (−3.58 − 3.00i)9-s + (2.19 + 0.469i)10-s + (−3.16 + 1.82i)11-s + (−4.98 + 2.41i)12-s + (−1.56 − 4.30i)13-s + (0.227 − 0.177i)14-s + (0.764 − 4.33i)15-s + (0.118 + 3.99i)16-s + (−3.79 + 3.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.926 − 0.375i)2-s + (−0.547 + 1.50i)3-s + (0.717 + 0.696i)4-s + (−0.699 + 0.123i)5-s + (1.07 − 1.18i)6-s + (−0.0385 + 0.0668i)7-s + (−0.403 − 0.915i)8-s + (−1.19 − 1.00i)9-s + (0.694 + 0.148i)10-s + (−0.953 + 0.550i)11-s + (−1.43 + 0.697i)12-s + (−0.434 − 1.19i)13-s + (0.0608 − 0.0474i)14-s + (0.197 − 1.11i)15-s + (0.0296 + 0.999i)16-s + (−0.919 + 0.771i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0313317 + 0.291518i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0313317 + 0.291518i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 + 0.531i)T \) |
| 19 | \( 1 + (-3.94 - 1.85i)T \) |
| good | 3 | \( 1 + (0.947 - 2.60i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (1.56 - 0.275i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.102 - 0.176i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.16 - 1.82i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.56 + 4.30i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (3.79 - 3.18i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.845 - 4.79i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.64 + 5.53i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.446 + 0.772i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 10.9iT - 37T^{2} \) |
| 41 | \( 1 + (-0.759 - 0.276i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (3.05 - 0.538i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.601 - 0.504i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (3.09 + 0.546i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.40 - 6.43i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.69 - 0.827i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.59 - 7.85i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.87 - 10.6i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-3.52 - 1.28i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (15.0 + 5.47i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (2.11 + 1.22i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 3.74i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-13.9 + 11.6i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.15007627505973237605346177798, −11.92263220022748819634413698642, −11.24275967439444465933723265577, −10.11335244789830843994527807090, −9.943498435978841726814597716575, −8.422858373937673449542113553952, −7.50635501259163893401972106119, −5.74066642363317455502836863612, −4.33918823997258476019229445739, −3.06218826522045995166885094618,
0.38770450154992167197343927392, 2.31680834397825912981295152665, 5.13743942925378456246189342086, 6.54098927653298599703925807166, 7.21799424085498657086528637588, 8.076421320285203342422264440171, 9.128016501000915888539032361458, 10.71799770945945404530057869214, 11.58035641814186195423125312167, 12.22830604515602021521633471309
