L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 2·11-s − 14-s + 16-s + 3·17-s − 8·19-s + 20-s − 2·22-s + 4·23-s − 4·25-s + 28-s + 5·29-s − 3·31-s − 32-s − 3·34-s + 35-s + 10·37-s + 8·38-s − 40-s + 41-s − 5·43-s + 2·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.83·19-s + 0.223·20-s − 0.426·22-s + 0.834·23-s − 4/5·25-s + 0.188·28-s + 0.928·29-s − 0.538·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s + 1.64·37-s + 1.29·38-s − 0.158·40-s + 0.156·41-s − 0.762·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.645107593\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.645107593\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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
−8.367685453432787752970629971919, −7.60244313029737587320284759588, −6.77260816609395551286192496066, −6.22521992764729152029152239897, −5.45821848324230659180784215026, −4.50956591206483164540917042476, −3.68629641490415248691631929400, −2.56209811383506646800247550920, −1.81335090619490852147633806567, −0.792448834555670241545210570180,
0.792448834555670241545210570180, 1.81335090619490852147633806567, 2.56209811383506646800247550920, 3.68629641490415248691631929400, 4.50956591206483164540917042476, 5.45821848324230659180784215026, 6.22521992764729152029152239897, 6.77260816609395551286192496066, 7.60244313029737587320284759588, 8.367685453432787752970629971919