| L(s) = 1 | − 5-s + 4·11-s − 4·13-s − 6·17-s − 6·19-s + 6·23-s + 25-s − 2·29-s + 8·31-s + 8·37-s − 4·43-s − 6·47-s − 7·49-s + 10·53-s − 4·55-s − 12·59-s + 6·61-s + 4·65-s + 8·67-s + 4·71-s + 14·73-s + 4·79-s − 4·83-s + 6·85-s − 4·89-s + 6·95-s + 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s − 1.10·13-s − 1.45·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 1.31·37-s − 0.609·43-s − 0.875·47-s − 49-s + 1.37·53-s − 0.539·55-s − 1.56·59-s + 0.768·61-s + 0.496·65-s + 0.977·67-s + 0.474·71-s + 1.63·73-s + 0.450·79-s − 0.439·83-s + 0.650·85-s − 0.423·89-s + 0.615·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.479743263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479743263\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−8.221784593047618370443600142588, −7.31234115119924924839519158305, −6.59955013140584688329875296949, −6.31922612991577539753934456878, −4.94021366721806789945391007986, −4.53528577234853491767187121078, −3.78396261931862396131954677074, −2.74258038034632227131235442262, −1.94258996435176496943866233694, −0.63273215052695375258695370179,
0.63273215052695375258695370179, 1.94258996435176496943866233694, 2.74258038034632227131235442262, 3.78396261931862396131954677074, 4.53528577234853491767187121078, 4.94021366721806789945391007986, 6.31922612991577539753934456878, 6.59955013140584688329875296949, 7.31234115119924924839519158305, 8.221784593047618370443600142588
