L(s) = 1 | + 3.34·3-s − 2.69·5-s + 0.531·7-s + 8.16·9-s − 3.11·11-s + 1.04·13-s − 9.01·15-s + 17-s + 7.95·19-s + 1.77·21-s + 3.29·23-s + 2.27·25-s + 17.2·27-s − 9.59·29-s − 4.60·31-s − 10.4·33-s − 1.43·35-s + 8.35·37-s + 3.50·39-s + 7.69·41-s − 0.0864·43-s − 22.0·45-s − 4.95·47-s − 6.71·49-s + 3.34·51-s − 0.349·53-s + 8.39·55-s + ⋯ |
L(s) = 1 | + 1.92·3-s − 1.20·5-s + 0.200·7-s + 2.72·9-s − 0.939·11-s + 0.290·13-s − 2.32·15-s + 0.242·17-s + 1.82·19-s + 0.387·21-s + 0.688·23-s + 0.454·25-s + 3.32·27-s − 1.78·29-s − 0.826·31-s − 1.81·33-s − 0.242·35-s + 1.37·37-s + 0.560·39-s + 1.20·41-s − 0.0131·43-s − 3.28·45-s − 0.722·47-s − 0.959·49-s + 0.467·51-s − 0.0479·53-s + 1.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.787416791\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.787416791\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 - 0.531T + 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 19 | \( 1 - 7.95T + 19T^{2} \) |
| 23 | \( 1 - 3.29T + 23T^{2} \) |
| 29 | \( 1 + 9.59T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 - 7.69T + 41T^{2} \) |
| 43 | \( 1 + 0.0864T + 43T^{2} \) |
| 47 | \( 1 + 4.95T + 47T^{2} \) |
| 53 | \( 1 + 0.349T + 53T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 7.81T + 67T^{2} \) |
| 71 | \( 1 + 7.42T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 7.74T + 83T^{2} \) |
| 89 | \( 1 + 2.13T + 89T^{2} \) |
| 97 | \( 1 + 0.643T + 97T^{2} \) |
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
−7.989495168483854542107898310626, −7.43770976512576845323530257900, −6.98798954826746412912139822530, −5.58537516871631504566052523335, −4.81056526437840449815979575200, −3.92888033729955071830865121971, −3.47130663148912036626154995158, −2.87261266835169904896056287208, −1.99158850818383168481636076673, −0.897020279220656045706438764354,
0.897020279220656045706438764354, 1.99158850818383168481636076673, 2.87261266835169904896056287208, 3.47130663148912036626154995158, 3.92888033729955071830865121971, 4.81056526437840449815979575200, 5.58537516871631504566052523335, 6.98798954826746412912139822530, 7.43770976512576845323530257900, 7.989495168483854542107898310626