| L(s) = 1 | − 4·5-s + 11-s − 13-s − 17-s − 2·19-s − 4·23-s + 11·25-s − 2·29-s − 6·37-s − 10·41-s + 6·43-s − 10·47-s − 7·49-s + 6·53-s − 4·55-s + 14·59-s + 8·61-s + 4·65-s − 16·67-s + 4·73-s − 12·79-s + 4·83-s + 4·85-s + 2·89-s + 8·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.301·11-s − 0.277·13-s − 0.242·17-s − 0.458·19-s − 0.834·23-s + 11/5·25-s − 0.371·29-s − 0.986·37-s − 1.56·41-s + 0.914·43-s − 1.45·47-s − 49-s + 0.824·53-s − 0.539·55-s + 1.82·59-s + 1.02·61-s + 0.496·65-s − 1.95·67-s + 0.468·73-s − 1.35·79-s + 0.439·83-s + 0.433·85-s + 0.211·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.01301643187071, −12.34695627904126, −12.01662789197312, −11.72636446055666, −11.30376327547160, −10.82221080738771, −10.37804198065127, −9.848007177765391, −9.365493844202997, −8.622698131368183, −8.398424108654255, −8.108994789042377, −7.413206478978765, −7.068911014779269, −6.710724267953759, −6.111038933135982, −5.379958485789551, −4.975619842318947, −4.284549884556623, −4.089067638126225, −3.469502845715475, −3.128744312032878, −2.336417532702818, −1.720880466857596, −0.9813121415301706, 0, 0,
0.9813121415301706, 1.720880466857596, 2.336417532702818, 3.128744312032878, 3.469502845715475, 4.089067638126225, 4.284549884556623, 4.975619842318947, 5.379958485789551, 6.111038933135982, 6.710724267953759, 7.068911014779269, 7.413206478978765, 8.108994789042377, 8.398424108654255, 8.622698131368183, 9.365493844202997, 9.848007177765391, 10.37804198065127, 10.82221080738771, 11.30376327547160, 11.72636446055666, 12.01662789197312, 12.34695627904126, 13.01301643187071
