L(s) = 1 | + 1.57·2-s − 3-s + 0.494·4-s − 1.92·5-s − 1.57·6-s − 2.95·7-s − 2.37·8-s + 9-s − 3.04·10-s − 0.760·11-s − 0.494·12-s − 6.68·13-s − 4.66·14-s + 1.92·15-s − 4.74·16-s + 17-s + 1.57·18-s − 0.0491·19-s − 0.953·20-s + 2.95·21-s − 1.20·22-s − 2.97·23-s + 2.37·24-s − 1.28·25-s − 10.5·26-s − 27-s − 1.45·28-s + ⋯ |
L(s) = 1 | + 1.11·2-s − 0.577·3-s + 0.247·4-s − 0.862·5-s − 0.644·6-s − 1.11·7-s − 0.840·8-s + 0.333·9-s − 0.962·10-s − 0.229·11-s − 0.142·12-s − 1.85·13-s − 1.24·14-s + 0.497·15-s − 1.18·16-s + 0.242·17-s + 0.372·18-s − 0.0112·19-s − 0.213·20-s + 0.643·21-s − 0.255·22-s − 0.619·23-s + 0.485·24-s − 0.256·25-s − 2.07·26-s − 0.192·27-s − 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6343852979\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6343852979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 - 1.57T + 2T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 + 2.95T + 7T^{2} \) |
| 11 | \( 1 + 0.760T + 11T^{2} \) |
| 13 | \( 1 + 6.68T + 13T^{2} \) |
| 19 | \( 1 + 0.0491T + 19T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 - 3.50T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 7.19T + 41T^{2} \) |
| 43 | \( 1 - 4.22T + 43T^{2} \) |
| 47 | \( 1 + 1.66T + 47T^{2} \) |
| 53 | \( 1 - 3.32T + 53T^{2} \) |
| 59 | \( 1 - 7.21T + 59T^{2} \) |
| 61 | \( 1 + 4.68T + 61T^{2} \) |
| 67 | \( 1 - 4.16T + 67T^{2} \) |
| 71 | \( 1 + 8.52T + 71T^{2} \) |
| 73 | \( 1 - 6.25T + 73T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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
−8.323873401048686854310942012395, −7.42351284973695384920382161092, −6.84889231211758795968683264593, −6.07901401149533847220251827149, −5.34205533421952160405863415809, −4.64776716210683503296703064450, −4.00062001375728113407084647295, −3.18393992258223700111391460091, −2.41497769994944023682669137988, −0.36831305702371742841162926913,
0.36831305702371742841162926913, 2.41497769994944023682669137988, 3.18393992258223700111391460091, 4.00062001375728113407084647295, 4.64776716210683503296703064450, 5.34205533421952160405863415809, 6.07901401149533847220251827149, 6.84889231211758795968683264593, 7.42351284973695384920382161092, 8.323873401048686854310942012395