| L(s) = 1 | + (8.70 + 2.27i)3-s + 28.9i·5-s − 2.36·7-s + (70.6 + 39.6i)9-s − 21.1i·11-s + 259.·13-s + (−65.8 + 251. i)15-s − 350. i·17-s + 308.·19-s + (−20.5 − 5.37i)21-s + 282. i·23-s − 211.·25-s + (524. + 506. i)27-s + 1.23e3i·29-s − 158.·31-s + ⋯ |
| L(s) = 1 | + (0.967 + 0.253i)3-s + 1.15i·5-s − 0.0481·7-s + (0.871 + 0.489i)9-s − 0.175i·11-s + 1.53·13-s + (−0.292 + 1.11i)15-s − 1.21i·17-s + 0.855·19-s + (−0.0466 − 0.0121i)21-s + 0.534i·23-s − 0.339·25-s + (0.719 + 0.694i)27-s + 1.47i·29-s − 0.164·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.193839616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.193839616\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-8.70 - 2.27i)T \) |
| good | 5 | \( 1 - 28.9iT - 625T^{2} \) |
| 7 | \( 1 + 2.36T + 2.40e3T^{2} \) |
| 11 | \( 1 + 21.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 259.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 350. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 308.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 282. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.23e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 158.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 890.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.78e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.44e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.40e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 580. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.75e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 205.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 610.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.01e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.54e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.30e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.44e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.87e3T + 8.85e7T^{2} \) |
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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
−10.92134721142840076235367525132, −9.900322936401038640740079033586, −9.141225823276650911546604511538, −8.113293152568174645725138015818, −7.22306848392959092222681249057, −6.34974371874500263134343810967, −4.91586558728205874937858659286, −3.40926326848573499211597647119, −3.02435603099318557113023482689, −1.42900850243718062830157344838,
0.890889777604678480609781961915, 1.88554549250929118000781693305, 3.48886377274214746016135083039, 4.34357708374291838659188008235, 5.69581337045637306540116396338, 6.79701771078311529235309599139, 8.202088311641829117656425090390, 8.440741502917163715606828842882, 9.410375436645118042573030088457, 10.30860635619990859808724199893
