L(s) = 1 | − 2i·2-s − 4·4-s + (3 + 4i)5-s + 8i·8-s − 9·9-s + (8 − 6i)10-s − 24i·13-s + 16·16-s + 16i·17-s + 18i·18-s + (−12 − 16i)20-s + (−7 + 24i)25-s − 48·26-s + 42·29-s − 32i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (0.600 + 0.800i)5-s + i·8-s − 9-s + (0.800 − 0.600i)10-s − 1.84i·13-s + 16-s + 0.941i·17-s + i·18-s + (−0.600 − 0.800i)20-s + (−0.280 + 0.959i)25-s − 1.84·26-s + 1.44·29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.741446 - 0.370723i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.741446 - 0.370723i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 5 | \( 1 + (-3 - 4i)T \) |
good | 3 | \( 1 + 9T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 24iT - 169T^{2} \) |
| 17 | \( 1 - 16iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 42T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 24iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18T + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 56iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 22T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 96iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 78T + 7.92e3T^{2} \) |
| 97 | \( 1 + 144iT - 9.40e3T^{2} \) |
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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
−17.96062590567325815104023523697, −17.33399256142956227684504519776, −15.02155578445027214922818563134, −13.91615602008035128526283404749, −12.62713949989021404985482988469, −11.02542479449272167448992573251, −10.09372891876765178042217088505, −8.324070771202873362775595250013, −5.70955971770895197519786291105, −2.98280952584571453573837281815,
4.87586661189760822228494762830, 6.50419717571286079875050761495, 8.520031008385668286043288259253, 9.547024869041151195264697437329, 11.90550244298984676775472499594, 13.59782577234706309083810554246, 14.34275859722948862111176454258, 16.12067171012368066821763565845, 16.87855723835633671836689974348, 17.94885027651970621775195863569