| L(s) = 1 | + 2·2-s + 4·4-s − 2·5-s − 7·7-s + 8·8-s − 4·10-s − 11·11-s + 26·13-s − 14·14-s + 16·16-s + 46·17-s − 48·19-s − 8·20-s − 22·22-s + 128·23-s − 121·25-s + 52·26-s − 28·28-s + 146·29-s − 128·31-s + 32·32-s + 92·34-s + 14·35-s − 26·37-s − 96·38-s − 16·40-s − 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.178·5-s − 0.377·7-s + 0.353·8-s − 0.126·10-s − 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.656·17-s − 0.579·19-s − 0.0894·20-s − 0.213·22-s + 1.16·23-s − 0.967·25-s + 0.392·26-s − 0.188·28-s + 0.934·29-s − 0.741·31-s + 0.176·32-s + 0.464·34-s + 0.0676·35-s − 0.115·37-s − 0.409·38-s − 0.0632·40-s − 0.0380·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.209818819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.209818819\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 46 T + p^{3} T^{2} \) |
| 19 | \( 1 + 48 T + p^{3} T^{2} \) |
| 23 | \( 1 - 128 T + p^{3} T^{2} \) |
| 29 | \( 1 - 146 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 26 T + p^{3} T^{2} \) |
| 41 | \( 1 + 10 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 544 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 48 T + p^{3} T^{2} \) |
| 61 | \( 1 - 466 T + p^{3} T^{2} \) |
| 67 | \( 1 - 516 T + p^{3} T^{2} \) |
| 71 | \( 1 - 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 754 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + 624 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1018 T + p^{3} T^{2} \) |
| 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
−9.203894298850877146025815253694, −8.311673116357176531058742209062, −7.46932386035642504146176784253, −6.64328751215213016611982411037, −5.83364346769184833016832634986, −5.03197101279504367249844564287, −3.99337525618611805006101131399, −3.24467506311083167341188943444, −2.17971905077352842321075225407, −0.802096479533121950393797408254,
0.802096479533121950393797408254, 2.17971905077352842321075225407, 3.24467506311083167341188943444, 3.99337525618611805006101131399, 5.03197101279504367249844564287, 5.83364346769184833016832634986, 6.64328751215213016611982411037, 7.46932386035642504146176784253, 8.311673116357176531058742209062, 9.203894298850877146025815253694
