| L(s) = 1 | + (2.23 − 2.66i)3-s + (1.62 − 0.590i)5-s + (−1.87 − 3.25i)7-s + (−0.538 − 3.05i)9-s + (2.58 − 4.47i)11-s + (3.02 + 3.60i)13-s + (2.05 − 5.64i)15-s + (−2.43 + 13.8i)17-s + (10.2 + 15.9i)19-s + (−12.8 − 2.27i)21-s + (−11.1 − 4.07i)23-s + (−16.8 + 14.1i)25-s + (17.7 + 10.2i)27-s + (−52.5 + 9.26i)29-s + (4.93 − 2.85i)31-s + ⋯ |
| L(s) = 1 | + (0.745 − 0.888i)3-s + (0.324 − 0.118i)5-s + (−0.268 − 0.465i)7-s + (−0.0598 − 0.339i)9-s + (0.234 − 0.406i)11-s + (0.232 + 0.277i)13-s + (0.136 − 0.376i)15-s + (−0.143 + 0.812i)17-s + (0.541 + 0.840i)19-s + (−0.613 − 0.108i)21-s + (−0.486 − 0.177i)23-s + (−0.674 + 0.566i)25-s + (0.657 + 0.379i)27-s + (−1.81 + 0.319i)29-s + (0.159 − 0.0919i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.42640 - 0.642673i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42640 - 0.642673i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-10.2 - 15.9i)T \) |
| good | 3 | \( 1 + (-2.23 + 2.66i)T + (-1.56 - 8.86i)T^{2} \) |
| 5 | \( 1 + (-1.62 + 0.590i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (1.87 + 3.25i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-2.58 + 4.47i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-3.02 - 3.60i)T + (-29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (2.43 - 13.8i)T + (-271. - 98.8i)T^{2} \) |
| 23 | \( 1 + (11.1 + 4.07i)T + (405. + 340. i)T^{2} \) |
| 29 | \( 1 + (52.5 - 9.26i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (-4.93 + 2.85i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 35.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-11.8 + 14.1i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-72.4 + 26.3i)T + (1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (6.56 + 37.2i)T + (-2.07e3 + 755. i)T^{2} \) |
| 53 | \( 1 + (-16.1 + 44.4i)T + (-2.15e3 - 1.80e3i)T^{2} \) |
| 59 | \( 1 + (86.2 + 15.2i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (19.7 + 7.19i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (46.1 - 8.13i)T + (4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (30.2 + 83.2i)T + (-3.86e3 + 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-57.7 - 48.4i)T + (925. + 5.24e3i)T^{2} \) |
| 79 | \( 1 + (10.5 - 12.5i)T + (-1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (17.5 + 30.4i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-18.1 - 21.6i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-39.0 - 6.87i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−13.84085446564356674964708182902, −13.30151978991230112714027951639, −12.20754808922933746815659293903, −10.77722720536900100935253772770, −9.440425885717309078980032240955, −8.246266485271863179058978240332, −7.22358376749354386008770657476, −5.86669578541877813213153495461, −3.68237233592570931394796446943, −1.75757949788587460176415086470,
2.73269415708788360547967080209, 4.25123753119227516274332028270, 5.88659139712576015321196305377, 7.56931698366834881079379646188, 9.194943835176409423606927352496, 9.543323265255129883145718970373, 10.93285974341759276237342687299, 12.25983043436617241239459027226, 13.57194878236563667891989474540, 14.49561899382332742305115999126
