| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s − 2·11-s − 2·13-s + 16-s + 4·17-s + 2·19-s + 3·20-s − 2·22-s − 2·23-s + 4·25-s − 2·26-s + 8·31-s + 32-s + 4·34-s + 3·37-s + 2·38-s + 3·40-s − 2·41-s + 6·43-s − 2·44-s − 2·46-s + 9·47-s − 7·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s − 0.603·11-s − 0.554·13-s + 1/4·16-s + 0.970·17-s + 0.458·19-s + 0.670·20-s − 0.426·22-s − 0.417·23-s + 4/5·25-s − 0.392·26-s + 1.43·31-s + 0.176·32-s + 0.685·34-s + 0.493·37-s + 0.324·38-s + 0.474·40-s − 0.312·41-s + 0.914·43-s − 0.301·44-s − 0.294·46-s + 1.31·47-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.037084086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.037084086\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.309679665222234265778832596810, −7.64447807082080030902822120847, −6.79049599562008690326245343638, −6.04897309284588006244814333442, −5.44913972699652728293766214307, −4.91751977639018926672851678728, −3.86791346292197773193617194413, −2.78774541069457029137201700543, −2.25696638679260588025447341411, −1.09248259894746013030137723471,
1.09248259894746013030137723471, 2.25696638679260588025447341411, 2.78774541069457029137201700543, 3.86791346292197773193617194413, 4.91751977639018926672851678728, 5.44913972699652728293766214307, 6.04897309284588006244814333442, 6.79049599562008690326245343638, 7.64447807082080030902822120847, 8.309679665222234265778832596810
