L(s) = 1 | − 2-s + 4-s + 3·5-s + 5·7-s − 8-s − 3·10-s + 11-s + 13-s − 5·14-s + 16-s + 2·19-s + 3·20-s − 22-s − 3·23-s + 4·25-s − 26-s + 5·28-s + 3·29-s + 8·31-s − 32-s + 15·35-s − 10·37-s − 2·38-s − 3·40-s + 3·41-s + 5·43-s + 44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 1.88·7-s − 0.353·8-s − 0.948·10-s + 0.301·11-s + 0.277·13-s − 1.33·14-s + 1/4·16-s + 0.458·19-s + 0.670·20-s − 0.213·22-s − 0.625·23-s + 4/5·25-s − 0.196·26-s + 0.944·28-s + 0.557·29-s + 1.43·31-s − 0.176·32-s + 2.53·35-s − 1.64·37-s − 0.324·38-s − 0.474·40-s + 0.468·41-s + 0.762·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.307779157\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.307779157\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−8.741536099403024989512024353554, −8.337271304742426211490506394748, −7.51904055050221723981686229374, −6.62831935974413748202292459727, −5.78957200773240421581851364343, −5.14141145022348185356564842937, −4.23959099978790744231346656734, −2.75021572913260944111096585983, −1.79775764616139241442996676187, −1.23949983076345421990973435055,
1.23949983076345421990973435055, 1.79775764616139241442996676187, 2.75021572913260944111096585983, 4.23959099978790744231346656734, 5.14141145022348185356564842937, 5.78957200773240421581851364343, 6.62831935974413748202292459727, 7.51904055050221723981686229374, 8.337271304742426211490506394748, 8.741536099403024989512024353554