L(s) = 1 | + (0.906 − 0.422i)2-s + (−0.199 + 0.284i)3-s + (0.642 − 0.766i)4-s + (−0.0603 + 0.342i)6-s + (−3.03 + 0.814i)7-s + (0.258 − 0.965i)8-s + (0.984 + 2.70i)9-s + (1.15 + 2.00i)11-s + (0.0898 + 0.335i)12-s + (−5.26 + 3.68i)13-s + (−2.40 + 2.02i)14-s + (−0.173 − 0.984i)16-s + (−0.101 − 0.217i)17-s + (2.03 + 2.03i)18-s + (−4.33 − 0.449i)19-s + ⋯ |
L(s) = 1 | + (0.640 − 0.298i)2-s + (−0.115 + 0.164i)3-s + (0.321 − 0.383i)4-s + (−0.0246 + 0.139i)6-s + (−1.14 + 0.307i)7-s + (0.0915 − 0.341i)8-s + (0.328 + 0.901i)9-s + (0.349 + 0.604i)11-s + (0.0259 + 0.0968i)12-s + (−1.46 + 1.02i)13-s + (−0.643 + 0.540i)14-s + (−0.0434 − 0.246i)16-s + (−0.0246 − 0.0527i)17-s + (0.479 + 0.479i)18-s + (−0.994 − 0.103i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.840907 + 0.969485i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.840907 + 0.969485i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.906 + 0.422i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.33 + 0.449i)T \) |
good | 3 | \( 1 + (0.199 - 0.284i)T + (-1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (3.03 - 0.814i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.15 - 2.00i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.26 - 3.68i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (0.101 + 0.217i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-3.03 + 0.265i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (9.33 - 3.39i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.73 - 4.46i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.25 - 2.25i)T - 37iT^{2} \) |
| 41 | \( 1 + (-11.4 + 2.01i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.0171 + 0.196i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.38 - 1.58i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.0547 - 0.625i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-9.92 - 3.61i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (7.07 + 5.93i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (4.90 - 10.5i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-1.44 - 1.72i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (12.5 + 8.79i)T + (24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (-1.43 - 8.13i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.739 - 2.76i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-2.79 + 15.8i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.340 + 0.158i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.24375236089025858069171090409, −9.611460719222809019512590948502, −8.886263540188911384702968016533, −7.35808065046747149323508081460, −6.89387892104024911103390648429, −5.86932105671155983594565257470, −4.79714651734726246128298422803, −4.20357242656026713199771441370, −2.84037615091398007420565066670, −1.95540064154849127573350126536,
0.45860931836515071073392167547, 2.55945786950872097052102169648, 3.52695001188387297535155801733, 4.37311542529394948928290657907, 5.69356825949216183877368860286, 6.28827022840251045850683418364, 7.12055855918885257924152060277, 7.84584427911776506714891660850, 9.114306955088370371044196412408, 9.765693754512389902249461604171