L(s) = 1 | + (−4.62 + 4.62i)5-s − 3.04·7-s + (9.15 + 9.15i)11-s + (5.78 + 5.78i)13-s − 17.6·17-s + (−1.15 + 1.15i)19-s − 3.45·23-s − 17.8i·25-s + (12.1 + 12.1i)29-s + 38.5i·31-s + (14.1 − 14.1i)35-s + (0.0972 − 0.0972i)37-s − 51.5i·41-s + (−1.70 − 1.70i)43-s − 24.1i·47-s + ⋯ |
L(s) = 1 | + (−0.925 + 0.925i)5-s − 0.435·7-s + (0.831 + 0.831i)11-s + (0.444 + 0.444i)13-s − 1.03·17-s + (−0.0606 + 0.0606i)19-s − 0.150·23-s − 0.712i·25-s + (0.420 + 0.420i)29-s + 1.24i·31-s + (0.403 − 0.403i)35-s + (0.00262 − 0.00262i)37-s − 1.25i·41-s + (−0.0395 − 0.0395i)43-s − 0.513i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3177347944\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3177347944\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (4.62 - 4.62i)T - 25iT^{2} \) |
| 7 | \( 1 + 3.04T + 49T^{2} \) |
| 11 | \( 1 + (-9.15 - 9.15i)T + 121iT^{2} \) |
| 13 | \( 1 + (-5.78 - 5.78i)T + 169iT^{2} \) |
| 17 | \( 1 + 17.6T + 289T^{2} \) |
| 19 | \( 1 + (1.15 - 1.15i)T - 361iT^{2} \) |
| 23 | \( 1 + 3.45T + 529T^{2} \) |
| 29 | \( 1 + (-12.1 - 12.1i)T + 841iT^{2} \) |
| 31 | \( 1 - 38.5iT - 961T^{2} \) |
| 37 | \( 1 + (-0.0972 + 0.0972i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 51.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (1.70 + 1.70i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 24.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-27.0 + 27.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (19.5 + 19.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (16.7 + 16.7i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (75.8 - 75.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 134.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 112. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 135. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (74.9 - 74.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 31.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31.5T + 9.40e3T^{2} \) |
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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
−10.22244962366017742352393676521, −9.157605151176470098170822589366, −8.548864595609908918381403007272, −7.31243385491973364082121820580, −6.90818143278528206215860408025, −6.16915887004128646873172708902, −4.70452146020924675718165008964, −3.89970897589967938317425331019, −3.07826589304159698531944584307, −1.75224893082775659043646071319,
0.10489654131579401405016466770, 1.18909179932744731685652238985, 2.88654953472402596806598487418, 3.98269768072816514404493531606, 4.54447308331471076089620390771, 5.84015731968942453836180048667, 6.50846777989529567153095024491, 7.66462869955510948282426340882, 8.412883755177944791366720949380, 8.960746107231182443367449050281