L(s) = 1 | + (0.173 + 0.984i)2-s + (2.49 − 2.09i)3-s + (−0.939 + 0.342i)4-s + (2.49 + 2.09i)6-s + (0.556 − 0.964i)7-s + (−0.5 − 0.866i)8-s + (1.31 − 7.46i)9-s + (0.0761 + 0.131i)11-s + (−1.62 + 2.81i)12-s + (2.66 + 2.23i)13-s + (1.04 + 0.380i)14-s + (0.766 − 0.642i)16-s + (0.0290 + 0.164i)17-s + 7.58·18-s + (−3.08 + 3.08i)19-s + ⋯ |
L(s) = 1 | + (0.122 + 0.696i)2-s + (1.43 − 1.20i)3-s + (−0.469 + 0.171i)4-s + (1.01 + 0.853i)6-s + (0.210 − 0.364i)7-s + (−0.176 − 0.306i)8-s + (0.438 − 2.48i)9-s + (0.0229 + 0.0397i)11-s + (−0.469 + 0.813i)12-s + (0.738 + 0.619i)13-s + (0.279 + 0.101i)14-s + (0.191 − 0.160i)16-s + (0.00704 + 0.0399i)17-s + 1.78·18-s + (−0.707 + 0.706i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.57615 - 0.790236i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.57615 - 0.790236i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.08 - 3.08i)T \) |
good | 3 | \( 1 + (-2.49 + 2.09i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.556 + 0.964i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0761 - 0.131i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.66 - 2.23i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0290 - 0.164i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.83 + 2.12i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.881 + 4.99i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.34 + 7.52i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.71T + 37T^{2} \) |
| 41 | \( 1 + (-3.70 + 3.10i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (9.50 + 3.46i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.46 - 8.28i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (13.3 - 4.87i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.35 - 13.3i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (4.92 - 1.79i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.450 + 2.55i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.70 - 0.622i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.79 + 6.53i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.00 + 3.35i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (3.03 - 5.26i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.18 - 7.70i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.16 - 12.2i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.461176399655069060275784940439, −8.887203746843459966220199787271, −8.064194804874182647819934039086, −7.62637658961609527697058433205, −6.62354196435477335175551832885, −6.15636844905231765405550490524, −4.45510021041447989029964083984, −3.59205241198614308602548134497, −2.41534242508037791952892094011, −1.17589325724473864689632461835,
1.78235865391217132716982170116, 3.05802075660222202109491130714, 3.42054187998191857342045572371, 4.70560433541157061071160550363, 5.18866491846906612732631766424, 6.82055132030692204311753406412, 8.254449726242306292621564329205, 8.544098526625589361976165220640, 9.311846662884502131526102819204, 10.07909640535747190161321001741