| L(s) = 1 | + (0.981 + 1.01i)2-s + (−2.24 − 0.477i)3-s + (−0.0748 + 1.99i)4-s + (1.04 + 1.81i)5-s + (−1.71 − 2.75i)6-s + (1.10 + 2.47i)7-s + (−2.10 + 1.88i)8-s + (2.06 + 0.921i)9-s + (−0.820 + 2.84i)10-s + (0.169 − 1.61i)11-s + (1.12 − 4.45i)12-s + (−1.67 − 1.51i)13-s + (−1.44 + 3.55i)14-s + (−1.48 − 4.57i)15-s + (−3.98 − 0.299i)16-s + (5.68 − 0.597i)17-s + ⋯ |
| L(s) = 1 | + (0.693 + 0.720i)2-s + (−1.29 − 0.275i)3-s + (−0.0374 + 0.999i)4-s + (0.468 + 0.811i)5-s + (−0.700 − 1.12i)6-s + (0.417 + 0.936i)7-s + (−0.745 + 0.666i)8-s + (0.689 + 0.307i)9-s + (−0.259 + 0.900i)10-s + (0.0511 − 0.486i)11-s + (0.323 − 1.28i)12-s + (−0.465 − 0.419i)13-s + (−0.385 + 0.950i)14-s + (−0.383 − 1.18i)15-s + (−0.997 − 0.0747i)16-s + (1.37 − 0.144i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.721090 + 0.816820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.721090 + 0.816820i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.981 - 1.01i)T \) |
| 31 | \( 1 + (-1.72 - 5.29i)T \) |
| good | 3 | \( 1 + (2.24 + 0.477i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (-1.04 - 1.81i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.10 - 2.47i)T + (-4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.169 + 1.61i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (1.67 + 1.51i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-5.68 + 0.597i)T + (16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-5.35 + 4.81i)T + (1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.21 - 0.881i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (10.1 + 3.28i)T + (23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-4.62 - 2.67i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.720 - 0.153i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (0.354 + 0.393i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.00753 + 0.00244i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.67 - 8.26i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-1.61 + 7.58i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 + (2.37 - 1.36i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.41 - 3.17i)T + (-47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-10.4 - 1.09i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.455 + 4.33i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (3.92 - 0.834i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (8.71 - 12.0i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.0620 + 0.0450i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.79307391948294885845026994232, −12.54899770877207412866579390715, −11.76970880238780556354037457365, −11.05471285146139712635383380107, −9.512011521651361398842533420350, −7.892829188107825778853780851793, −6.78034924308980819581999036969, −5.70838471326797343266722758965, −5.21947604763773504233228102712, −2.96798551925233552920365514025,
1.30886552522166708519421604617, 4.00917915733046405642181270883, 5.15011280980939559982844766698, 5.80749895718867368211415296652, 7.45241938443674255085698429128, 9.554957988934012214306364420100, 10.19078619826000848199057309949, 11.27840346354314717023722499653, 12.05661442348169627954718371362, 12.85926320562592510438662599479
