| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 1.21i·5-s − 2.82·8-s − 1.71i·10-s − 6.34·11-s + 8.15i·13-s + 4.00·16-s + 6.88i·17-s − 30.4i·19-s + 2.42i·20-s + 8.97·22-s − 36.9·23-s + 23.5·25-s − 11.5i·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.242i·5-s − 0.353·8-s − 0.171i·10-s − 0.576·11-s + 0.627i·13-s + 0.250·16-s + 0.405i·17-s − 1.60i·19-s + 0.121i·20-s + 0.407·22-s − 1.60·23-s + 0.941·25-s − 0.443i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6918747253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6918747253\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.21iT - 25T^{2} \) |
| 11 | \( 1 + 6.34T + 121T^{2} \) |
| 13 | \( 1 - 8.15iT - 169T^{2} \) |
| 17 | \( 1 - 6.88iT - 289T^{2} \) |
| 19 | \( 1 + 30.4iT - 361T^{2} \) |
| 23 | \( 1 + 36.9T + 529T^{2} \) |
| 29 | \( 1 - 34.3T + 841T^{2} \) |
| 31 | \( 1 + 17.8iT - 961T^{2} \) |
| 37 | \( 1 + 36.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 32.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 36.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 1.94iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 78.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 30.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 48.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 70.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 30.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 66.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 155. iT - 9.40e3T^{2} \) |
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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
−9.790068355485657374484691089373, −8.767779887204886296988180080777, −8.213313380083784815541772120596, −7.10124793147013903716775284135, −6.55347110516277650381593716300, −5.39122276620086293579905845160, −4.28350013639424029067318197010, −2.95206268756440507799760779502, −1.93478905707711902997454443835, −0.30548638233694494270405231491,
1.19214644824302628491362526929, 2.51599179126604180681276097392, 3.67053498227267800778549335741, 5.00557678304510097978112320939, 5.90303315147217947219512000544, 6.85607364898595694231011728511, 7.979112036764094103527690015236, 8.286080900865313119410206753023, 9.359695562979651133663570772631, 10.38584829908277770034877319006
