| L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.142 − 0.989i)3-s + (0.415 + 0.909i)4-s + (1.61 + 0.474i)5-s + (0.415 − 0.909i)6-s + (0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)9-s + (1.10 + 1.27i)10-s + (2.58 − 1.66i)11-s + (0.841 − 0.540i)12-s + (−0.336 − 0.387i)13-s + (0.959 − 0.281i)14-s + (0.239 − 1.66i)15-s + (−0.654 + 0.755i)16-s + (0.715 − 1.56i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (0.721 + 0.211i)5-s + (0.169 − 0.371i)6-s + (0.247 − 0.285i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.348 + 0.402i)10-s + (0.779 − 0.500i)11-s + (0.242 − 0.156i)12-s + (−0.0932 − 0.107i)13-s + (0.256 − 0.0752i)14-s + (0.0618 − 0.429i)15-s + (−0.163 + 0.188i)16-s + (0.173 − 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.62633 + 0.0812265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.62633 + 0.0812265i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-3.97 + 2.67i)T \) |
| good | 5 | \( 1 + (-1.61 - 0.474i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-2.58 + 1.66i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.336 + 0.387i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.715 + 1.56i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.77 - 6.08i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.06 - 4.51i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.39 + 9.69i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-7.08 + 2.08i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.95 - 0.867i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.55 - 10.7i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + (4.33 - 5.00i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.74 - 2.01i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.01 + 7.09i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.33 + 2.14i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-2.49 - 1.60i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.40 + 5.25i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-4.25 - 4.91i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.867 + 0.254i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.44 + 10.0i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (9.35 + 2.74i)T + (81.6 + 52.4i)T^{2} \) |
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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
−9.945485926789459905563457513837, −9.209919803435948567194355915951, −8.022920780124578791258045175883, −7.49396732776953223673498143266, −6.30204597905486951743936845045, −6.01395997369167578585065094715, −4.90006091480479804239832300740, −3.74704765910515592585740880685, −2.62708571247102880160945875362, −1.30216757414511098351793687724,
1.40621135076618486855794110331, 2.63104669659323647506848429641, 3.76647378901073909725732444012, 4.82211186430914026339529506636, 5.39554867529649628006799101624, 6.38906055718000448827435215426, 7.28723585361094122758800215086, 8.675065023183406262481696420136, 9.428337250915469121336735076537, 9.896970150593072162056471437555
