| L(s) = 1 | + 3-s + 0.537·5-s + 7-s + 9-s + 0.264·11-s + 3.16·13-s + 0.537·15-s − 1.70·17-s + 1.73·19-s + 21-s + 23-s − 4.71·25-s + 27-s − 2.62·29-s − 2.62·31-s + 0.264·33-s + 0.537·35-s + 3.70·37-s + 3.16·39-s + 0.659·41-s + 7.71·43-s + 0.537·45-s + 2.89·47-s + 49-s − 1.70·51-s + 11.2·53-s + 0.142·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.240·5-s + 0.377·7-s + 0.333·9-s + 0.0796·11-s + 0.877·13-s + 0.138·15-s − 0.412·17-s + 0.398·19-s + 0.218·21-s + 0.208·23-s − 0.942·25-s + 0.192·27-s − 0.487·29-s − 0.471·31-s + 0.0460·33-s + 0.0909·35-s + 0.608·37-s + 0.506·39-s + 0.103·41-s + 1.17·43-s + 0.0801·45-s + 0.421·47-s + 0.142·49-s − 0.238·51-s + 1.54·53-s + 0.0191·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.076620160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.076620160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 0.537T + 5T^{2} \) |
| 11 | \( 1 - 0.264T + 11T^{2} \) |
| 13 | \( 1 - 3.16T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 - 0.659T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 7.97T + 59T^{2} \) |
| 61 | \( 1 + 6.60T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 7.17T + 73T^{2} \) |
| 79 | \( 1 + 7.50T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 6.08T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.76940871384075975458183342085, −7.39400338625324874345305799440, −6.44404357263260369432873814099, −5.81283003195828046282008302059, −5.07713464493568397910024316506, −4.10245903489874841455440257350, −3.64867155528023568742995887472, −2.58648872868268205564256625564, −1.87880254153315341627424205900, −0.878018215678570523990908579354,
0.878018215678570523990908579354, 1.87880254153315341627424205900, 2.58648872868268205564256625564, 3.64867155528023568742995887472, 4.10245903489874841455440257350, 5.07713464493568397910024316506, 5.81283003195828046282008302059, 6.44404357263260369432873814099, 7.39400338625324874345305799440, 7.76940871384075975458183342085
