L(s) = 1 | + (1.22 + 1.22i)3-s + (−4.78 − 4.78i)5-s + 10.3·7-s + 2.99i·9-s + (0.526 − 0.526i)11-s + (17.2 − 17.2i)13-s − 11.7i·15-s + 4.71·17-s + (2.53 + 2.53i)19-s + (12.6 + 12.6i)21-s + 12.5·23-s + 20.8i·25-s + (−3.67 + 3.67i)27-s + (−2.19 + 2.19i)29-s + 28.0i·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.957 − 0.957i)5-s + 1.47·7-s + 0.333i·9-s + (0.0478 − 0.0478i)11-s + (1.32 − 1.32i)13-s − 0.781i·15-s + 0.277·17-s + (0.133 + 0.133i)19-s + (0.602 + 0.602i)21-s + 0.547·23-s + 0.834i·25-s + (−0.136 + 0.136i)27-s + (−0.0757 + 0.0757i)29-s + 0.904i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.70965 - 0.352018i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70965 - 0.352018i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
good | 5 | \( 1 + (4.78 + 4.78i)T + 25iT^{2} \) |
| 7 | \( 1 - 10.3T + 49T^{2} \) |
| 11 | \( 1 + (-0.526 + 0.526i)T - 121iT^{2} \) |
| 13 | \( 1 + (-17.2 + 17.2i)T - 169iT^{2} \) |
| 17 | \( 1 - 4.71T + 289T^{2} \) |
| 19 | \( 1 + (-2.53 - 2.53i)T + 361iT^{2} \) |
| 23 | \( 1 - 12.5T + 529T^{2} \) |
| 29 | \( 1 + (2.19 - 2.19i)T - 841iT^{2} \) |
| 31 | \( 1 - 28.0iT - 961T^{2} \) |
| 37 | \( 1 + (32.1 + 32.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 23.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (4.79 - 4.79i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 39.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (27.9 + 27.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (79.8 - 79.8i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (36.7 - 36.7i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-10.9 - 10.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 52.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 67.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 56.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-58.3 - 58.3i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 60.9T + 9.40e3T^{2} \) |
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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
−12.18415593254377227260947353223, −11.22803959858857303714146698019, −10.47266861774124663090502157174, −8.820561130594312063907243403119, −8.342051612761524327883983924812, −7.56708596576351764811886720817, −5.51565548405814255013620856131, −4.60906651651711544847946562674, −3.45730028664858211446216825770, −1.21514445701096959665337310196,
1.67881635611071963868465396561, 3.41136548539022760971683745201, 4.56976145684477704101835818951, 6.35203512508223981699569409065, 7.40847357441300929315128456169, 8.146222182248432338632068491330, 9.129612048650145158472619611708, 10.83975628255485267128524313165, 11.33126947332046220642382051660, 12.09066492418728341252417607028