L(s) = 1 | − 8i·5-s + 24i·13-s − 30·17-s − 39·25-s + 40i·29-s − 24i·37-s − 18·41-s + 49·49-s + 56i·53-s + 120i·61-s + 192·65-s + 110·73-s + 240i·85-s − 78·89-s − 130·97-s + ⋯ |
L(s) = 1 | − 1.60i·5-s + 1.84i·13-s − 1.76·17-s − 1.56·25-s + 1.37i·29-s − 0.648i·37-s − 0.439·41-s + 0.999·49-s + 1.05i·53-s + 1.96i·61-s + 2.95·65-s + 1.50·73-s + 2.82i·85-s − 0.876·89-s − 1.34·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7952995811\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7952995811\) |
\(L(2)\) |
|
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 | \( 1 + 8iT - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 24iT - 169T^{2} \) |
| 17 | \( 1 + 30T + 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 40iT - 841T^{2} \) |
| 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 | \( 1 - 120iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 110T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 78T + 7.92e3T^{2} \) |
| 97 | \( 1 + 130T + 9.40e3T^{2} \) |
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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
−9.429589192599625196528852247727, −8.947412239860135317690157458352, −8.553712908357946037454445823122, −7.27239906329365288191992880793, −6.52076613214452897279303420803, −5.40304349924919107735329091352, −4.53456956091277941066100927246, −4.04099505427912534838183770285, −2.24595443894314882882760091766, −1.27631972189360007629408374547,
0.23927286703211987287071772735, 2.24300894325048761371351860277, 2.98829560410265368510239700151, 3.94296339704737778707105454369, 5.21284316672106096258114447173, 6.24682051394766786845046640549, 6.79589879245493768838905554065, 7.73816542994360663352345466260, 8.400380869890373878263185751725, 9.637129277635446993759658749884