L(s) = 1 | + 2.73·3-s + 5-s − 0.732·7-s + 4.46·9-s + 2·11-s − 3.46·13-s + 2.73·15-s + 3.46·17-s + 0.535·19-s − 2·21-s + 6.19·23-s + 25-s + 3.99·27-s + 6.92·29-s − 5.46·31-s + 5.46·33-s − 0.732·35-s + 2·37-s − 9.46·39-s − 1.46·41-s + 5.26·43-s + 4.46·45-s + 3.26·47-s − 6.46·49-s + 9.46·51-s − 11.4·53-s + 2·55-s + ⋯ |
L(s) = 1 | + 1.57·3-s + 0.447·5-s − 0.276·7-s + 1.48·9-s + 0.603·11-s − 0.960·13-s + 0.705·15-s + 0.840·17-s + 0.122·19-s − 0.436·21-s + 1.29·23-s + 0.200·25-s + 0.769·27-s + 1.28·29-s − 0.981·31-s + 0.951·33-s − 0.123·35-s + 0.328·37-s − 1.51·39-s − 0.228·41-s + 0.803·43-s + 0.665·45-s + 0.476·47-s − 0.923·49-s + 1.32·51-s − 1.57·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.120079751\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.120079751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.535T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 - 5.26T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 7.46T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 5.46T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.505787242908214611883146459689, −8.994570710974860012793157397802, −8.119786568776193441008436788120, −7.35522326405815039229326843306, −6.61567147189225814031160257017, −5.38047197900176002481020886461, −4.33615390727861181813506908219, −3.24330185652132838407459964870, −2.63435167007807001565409260723, −1.41539907066851654198387062095,
1.41539907066851654198387062095, 2.63435167007807001565409260723, 3.24330185652132838407459964870, 4.33615390727861181813506908219, 5.38047197900176002481020886461, 6.61567147189225814031160257017, 7.35522326405815039229326843306, 8.119786568776193441008436788120, 8.994570710974860012793157397802, 9.505787242908214611883146459689