L(s) = 1 | + (0.253 + 1.44i)2-s + (−0.130 + 0.0473i)4-s + (−0.939 − 0.342i)5-s + (−2.03 + 3.52i)7-s + (1.36 + 2.35i)8-s + (0.253 − 1.44i)10-s + (−0.310 − 0.537i)11-s + (−3.90 − 3.27i)13-s + (−5.59 − 2.03i)14-s + (−3.26 + 2.73i)16-s + (0.0462 + 0.262i)17-s + (0.399 + 4.34i)19-s + 0.138·20-s + (0.695 − 0.583i)22-s + (−5.48 + 1.99i)23-s + ⋯ |
L(s) = 1 | + (0.179 + 1.01i)2-s + (−0.0650 + 0.0236i)4-s + (−0.420 − 0.152i)5-s + (−0.769 + 1.33i)7-s + (0.481 + 0.833i)8-s + (0.0802 − 0.455i)10-s + (−0.0936 − 0.162i)11-s + (−1.08 − 0.908i)13-s + (−1.49 − 0.544i)14-s + (−0.815 + 0.684i)16-s + (0.0112 + 0.0636i)17-s + (0.0917 + 0.995i)19-s + 0.0309·20-s + (0.148 − 0.124i)22-s + (−1.14 + 0.416i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.132590 - 0.933364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.132590 - 0.933364i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.399 - 4.34i)T \) |
good | 2 | \( 1 + (-0.253 - 1.44i)T + (-1.87 + 0.684i)T^{2} \) |
| 7 | \( 1 + (2.03 - 3.52i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.310 + 0.537i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.90 + 3.27i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0462 - 0.262i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (5.48 - 1.99i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.708 - 4.01i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.83T + 37T^{2} \) |
| 41 | \( 1 + (-3.43 + 2.88i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.69 - 0.615i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (2.00 - 11.3i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (2.37 - 0.862i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.154 + 0.876i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.03 - 0.742i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.44 - 13.8i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (1.54 + 0.563i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.37 + 1.15i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.94 + 3.30i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.89 + 11.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.000572 - 0.000480i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.80 + 10.2i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.49862070842311967449327127584, −9.688177404613218361458163970640, −8.707935483901401853804079828523, −7.919908933845696405761055749506, −7.29412033298037884104163915587, −6.03694927534092159768484222630, −5.74606989650517085665362073216, −4.73438140428339983065992547414, −3.27321832557500147001034224671, −2.17043271663824419284206803104,
0.39669686313096904331764395343, 2.05541797026759130019245814013, 3.18048940255799940857460915656, 4.04867678771926116749962573868, 4.77288077982781387260565973382, 6.64524245240975081807501250922, 6.98740157726695341490994258374, 7.86412566821539197274733236762, 9.261672355669199292048076777514, 10.05400310883163219755534947886