L(s) = 1 | + (−0.997 + 0.0747i)2-s + (0.988 − 0.149i)4-s + (−0.974 + 0.222i)8-s + (0.0747 + 0.997i)11-s + (0.563 + 0.826i)13-s + (0.955 − 0.294i)16-s + (−0.781 − 0.623i)17-s + 19-s + (−0.149 − 0.988i)22-s + (0.149 + 0.988i)23-s + (−0.623 − 0.781i)26-s + (0.988 + 0.149i)29-s + (0.5 − 0.866i)31-s + (−0.930 + 0.365i)32-s + (0.826 + 0.563i)34-s + ⋯ |
L(s) = 1 | + (−0.997 + 0.0747i)2-s + (0.988 − 0.149i)4-s + (−0.974 + 0.222i)8-s + (0.0747 + 0.997i)11-s + (0.563 + 0.826i)13-s + (0.955 − 0.294i)16-s + (−0.781 − 0.623i)17-s + 19-s + (−0.149 − 0.988i)22-s + (0.149 + 0.988i)23-s + (−0.623 − 0.781i)26-s + (0.988 + 0.149i)29-s + (0.5 − 0.866i)31-s + (−0.930 + 0.365i)32-s + (0.826 + 0.563i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9519249746 + 0.4803745501i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9519249746 + 0.4803745501i\) |
\(L(1)\) |
\(\approx\) |
\(0.7544839584 + 0.1275317907i\) |
\(L(1)\) |
\(\approx\) |
\(0.7544839584 + 0.1275317907i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.997 + 0.0747i)T \) |
| 11 | \( 1 + (0.0747 + 0.997i)T \) |
| 13 | \( 1 + (0.563 + 0.826i)T \) |
| 17 | \( 1 + (-0.781 - 0.623i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.149 + 0.988i)T \) |
| 29 | \( 1 + (0.988 + 0.149i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.781 + 0.623i)T \) |
| 41 | \( 1 + (-0.955 - 0.294i)T \) |
| 43 | \( 1 + (-0.294 - 0.955i)T \) |
| 47 | \( 1 + (0.997 - 0.0747i)T \) |
| 53 | \( 1 + (0.781 - 0.623i)T \) |
| 59 | \( 1 + (-0.733 - 0.680i)T \) |
| 61 | \( 1 + (0.988 + 0.149i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.433 - 0.900i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.563 + 0.826i)T \) |
| 89 | \( 1 + (-0.900 + 0.433i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.71263771884806308388252031493, −18.77985006970218680247763897026, −18.1866651495227225190805784486, −17.61803639828441916980865470933, −16.78201977082819862880605033444, −16.09228502691531881788321068586, −15.58036027450425039427499782653, −14.73570557124699174304632700118, −13.79467767263750239180818212816, −13.00189351292017420706255052648, −12.13485465053983334481043939828, −11.38917531297603039158486813858, −10.65189593605185958941171713783, −10.19914212532586713827571601543, −9.11531697345842309651757341764, −8.501668921234822742153682847468, −7.99752515152982363273029741301, −6.97105529568863273269573674565, −6.22441761348440220917897670795, −5.5821819264693898716894685157, −4.32037096554052987134674771464, −3.19148685439955370314763207221, −2.69551401044577560303190810897, −1.40338424413875280973439870068, −0.64508812461601443976758226951,
0.93170025539440243475892225440, 1.84896799089472998951655680294, 2.63476586654755178508277277135, 3.707695143901835356056751646164, 4.74511243275691031789410583085, 5.67313569065170782017318334098, 6.73188569318760894807149915441, 7.0869344479623451350440825480, 8.00424693511327761932935983385, 8.822635901325519138640211437707, 9.54814739354596678082627076648, 10.01140301741501745824651554797, 11.0497816073555201314841483421, 11.68735060068276414009810858627, 12.19725051780771462129234937671, 13.42623485703908246062694586612, 14.01210548199295727038474818674, 15.17583707871833639067462808458, 15.53523619250263755287965395371, 16.32160129198285762431329986522, 17.071986914545652354136478582118, 17.75982594219190346753553604027, 18.33284661276233751410131258459, 18.98668987406192676309340235397, 19.87548236834703207580347234636