| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 4·11-s − 2·13-s + 16-s + 4·19-s − 2·20-s + 4·22-s − 25-s + 2·26-s − 10·29-s − 8·31-s − 32-s + 2·37-s − 4·38-s + 2·40-s + 10·41-s + 12·43-s − 4·44-s − 7·49-s + 50-s − 2·52-s − 6·53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.554·13-s + 1/4·16-s + 0.917·19-s − 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.392·26-s − 1.85·29-s − 1.43·31-s − 0.176·32-s + 0.328·37-s − 0.648·38-s + 0.316·40-s + 1.56·41-s + 1.82·43-s − 0.603·44-s − 49-s + 0.141·50-s − 0.277·52-s − 0.824·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5564611441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5564611441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.83930938764568140317765138944, −7.66986248445833156449605195643, −7.26300963540780008642691013326, −5.99765721904693160562602826795, −5.43472822041938456358161596847, −4.49188877224717986660432740122, −3.58617728841406145897914485292, −2.78267662915854226444354093506, −1.83426448749876956919734046077, −0.43196243578335716246083514587,
0.43196243578335716246083514587, 1.83426448749876956919734046077, 2.78267662915854226444354093506, 3.58617728841406145897914485292, 4.49188877224717986660432740122, 5.43472822041938456358161596847, 5.99765721904693160562602826795, 7.26300963540780008642691013326, 7.66986248445833156449605195643, 7.83930938764568140317765138944
