L(s) = 1 | − 7-s − 11-s + 13-s + 17-s + 4·19-s + 3·23-s − 8·29-s − 4·31-s − 3·37-s + 9·41-s + 8·43-s − 10·47-s − 6·49-s − 53-s + 4·59-s + 11·61-s + 4·67-s + 71-s + 14·73-s + 77-s + 79-s + 6·83-s + 15·89-s − 91-s − 15·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 0.301·11-s + 0.277·13-s + 0.242·17-s + 0.917·19-s + 0.625·23-s − 1.48·29-s − 0.718·31-s − 0.493·37-s + 1.40·41-s + 1.21·43-s − 1.45·47-s − 6/7·49-s − 0.137·53-s + 0.520·59-s + 1.40·61-s + 0.488·67-s + 0.118·71-s + 1.63·73-s + 0.113·77-s + 0.112·79-s + 0.658·83-s + 1.58·89-s − 0.104·91-s − 1.52·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.167803894\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167803894\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−13.07482946414159, −12.75973266312547, −12.28975819565185, −11.60294128308074, −11.23226937963299, −10.86949987687779, −10.31492278644306, −9.693285856990466, −9.313386912373166, −9.075658824757697, −8.246893887215608, −7.802597975593023, −7.458084971622946, −6.780270297958821, −6.445664355040858, −5.659803410072141, −5.365051449696048, −4.905744539420159, −4.035756465988829, −3.643992368003479, −3.147840755672319, −2.472566005805292, −1.876512826208299, −1.112300505501237, −0.4566819503074388,
0.4566819503074388, 1.112300505501237, 1.876512826208299, 2.472566005805292, 3.147840755672319, 3.643992368003479, 4.035756465988829, 4.905744539420159, 5.365051449696048, 5.659803410072141, 6.445664355040858, 6.780270297958821, 7.458084971622946, 7.802597975593023, 8.246893887215608, 9.075658824757697, 9.313386912373166, 9.693285856990466, 10.31492278644306, 10.86949987687779, 11.23226937963299, 11.60294128308074, 12.28975819565185, 12.75973266312547, 13.07482946414159