| L(s) = 1 | + (0.0503 + 0.145i)2-s + (−0.458 + 0.888i)3-s + (1.55 − 1.22i)4-s + (−3.87 − 0.370i)5-s + (−0.152 − 0.0219i)6-s + (2.22 + 1.43i)7-s + (0.515 + 0.331i)8-s + (−0.580 − 0.814i)9-s + (−0.141 − 0.582i)10-s + (3.50 + 1.21i)11-s + (0.374 + 1.94i)12-s + (−1.12 + 3.84i)13-s + (−0.0972 + 0.395i)14-s + (2.10 − 3.27i)15-s + (0.909 − 3.74i)16-s + (−0.758 + 0.303i)17-s + ⋯ |
| L(s) = 1 | + (0.0356 + 0.102i)2-s + (−0.264 + 0.513i)3-s + (0.776 − 0.610i)4-s + (−1.73 − 0.165i)5-s + (−0.0622 − 0.00894i)6-s + (0.839 + 0.543i)7-s + (0.182 + 0.117i)8-s + (−0.193 − 0.271i)9-s + (−0.0446 − 0.184i)10-s + (1.05 + 0.365i)11-s + (0.107 + 0.560i)12-s + (−0.313 + 1.06i)13-s + (−0.0259 + 0.105i)14-s + (0.543 − 0.845i)15-s + (0.227 − 0.937i)16-s + (−0.183 + 0.0736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21146 + 0.555234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21146 + 0.555234i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.458 - 0.888i)T \) |
| 7 | \( 1 + (-2.22 - 1.43i)T \) |
| 23 | \( 1 + (-3.94 - 2.72i)T \) |
| good | 2 | \( 1 + (-0.0503 - 0.145i)T + (-1.57 + 1.23i)T^{2} \) |
| 5 | \( 1 + (3.87 + 0.370i)T + (4.90 + 0.946i)T^{2} \) |
| 11 | \( 1 + (-3.50 - 1.21i)T + (8.64 + 6.79i)T^{2} \) |
| 13 | \( 1 + (1.12 - 3.84i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.758 - 0.303i)T + (12.3 - 11.7i)T^{2} \) |
| 19 | \( 1 + (-5.11 - 2.04i)T + (13.7 + 13.1i)T^{2} \) |
| 29 | \( 1 + (0.161 - 1.12i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.28 - 0.346i)T + (30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (7.73 - 5.51i)T + (12.1 - 34.9i)T^{2} \) |
| 41 | \( 1 + (2.88 - 1.31i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (5.19 + 8.08i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (1.73 + 0.999i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.24 - 3.40i)T + (-2.52 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-11.3 + 2.75i)T + (52.4 - 27.0i)T^{2} \) |
| 61 | \( 1 + (2.78 - 1.43i)T + (35.3 - 49.6i)T^{2} \) |
| 67 | \( 1 + (0.171 - 0.888i)T + (-62.2 - 24.9i)T^{2} \) |
| 71 | \( 1 + (-2.64 + 3.04i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (10.3 + 13.1i)T + (-17.2 + 70.9i)T^{2} \) |
| 79 | \( 1 + (-5.20 - 5.46i)T + (-3.75 + 78.9i)T^{2} \) |
| 83 | \( 1 + (2.52 - 5.52i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (0.802 + 16.8i)T + (-88.5 + 8.45i)T^{2} \) |
| 97 | \( 1 + (-0.426 - 0.933i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.51255565207778173268295033586, −10.38994839432076910088957336190, −9.287930366255150677991130108897, −8.398129843227444333729697918662, −7.33410449559989024715358492706, −6.66090538542309857183053049783, −5.20911314069520669377949761560, −4.50803015110241841335635280967, −3.33736431217693534939159957745, −1.46888746144569403116170900983,
0.971674800228379047040633373394, 2.94411977406822644951689988785, 3.85648870456629915008686128083, 4.98751442016150545691935736144, 6.66497291677751838200185432048, 7.27812147145405914116218677284, 7.962400050619874322673417434268, 8.595175516145934931612530639187, 10.39955209791597248531465586830, 11.32823274689066385557794982745
