| L(s) = 1 | + 16.9i·3-s + (12.3 − 54.5i)5-s − 211. i·7-s − 45.4·9-s − 520.·11-s + 732. i·13-s + (925. + 210. i)15-s + 2.26e3i·17-s − 2.03e3·19-s + 3.59e3·21-s + 974. i·23-s + (−2.81e3 − 1.35e3i)25-s + 3.35e3i·27-s − 5.27e3·29-s − 2.00e3·31-s + ⋯ |
| L(s) = 1 | + 1.08i·3-s + (0.221 − 0.975i)5-s − 1.63i·7-s − 0.186·9-s − 1.29·11-s + 1.20i·13-s + (1.06 + 0.241i)15-s + 1.90i·17-s − 1.29·19-s + 1.77·21-s + 0.384i·23-s + (−0.901 − 0.432i)25-s + 0.885i·27-s − 1.16·29-s − 0.374·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4042544095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4042544095\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-12.3 + 54.5i)T \) |
| good | 3 | \( 1 - 16.9iT - 243T^{2} \) |
| 7 | \( 1 + 211. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 520.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 732. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 2.26e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 974. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.65e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.07e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.33e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.74e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.89e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.73e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.09e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.82e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.41e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.62e4iT - 8.58e9T^{2} \) |
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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
−12.77711326228984395404642886504, −10.97751744849409122348141410937, −10.47328551680601193848749850494, −9.571266401984724021346031683707, −8.477753074084656406635522427319, −7.32922601137002557798782179347, −5.75905032292405166167854194936, −4.31897881935346840321325090156, −4.05377929540918045933457868016, −1.65181339006130151791092158244,
0.12383073429486906305957852043, 2.27136148484139237088018510322, 2.78394668976824701199523849028, 5.27454154667343071928992701627, 6.15394390162109296103898057967, 7.32550416354775518444547907672, 8.126104625152492912070163576547, 9.448584665583506375330431944154, 10.63974530175612674202552180707, 11.63891652430829769825628068793
