| L(s) = 1 | − 1.51·3-s + 6.33·5-s − 1.77·7-s − 24.7·9-s − 48.2·11-s + 6.22·13-s − 9.58·15-s + 34.0·17-s + 54.3·19-s + 2.69·21-s + 189.·23-s − 84.9·25-s + 78.2·27-s + 29·29-s + 37.5·31-s + 73.0·33-s − 11.2·35-s + 61.4·37-s − 9.42·39-s + 164.·41-s + 439.·43-s − 156.·45-s − 165.·47-s − 339.·49-s − 51.4·51-s − 572.·53-s − 305.·55-s + ⋯ |
| L(s) = 1 | − 0.291·3-s + 0.566·5-s − 0.0960·7-s − 0.915·9-s − 1.32·11-s + 0.132·13-s − 0.164·15-s + 0.485·17-s + 0.656·19-s + 0.0279·21-s + 1.71·23-s − 0.679·25-s + 0.557·27-s + 0.185·29-s + 0.217·31-s + 0.385·33-s − 0.0543·35-s + 0.273·37-s − 0.0386·39-s + 0.627·41-s + 1.56·43-s − 0.518·45-s − 0.514·47-s − 0.990·49-s − 0.141·51-s − 1.48·53-s − 0.749·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 1.51T + 27T^{2} \) |
| 5 | \( 1 - 6.33T + 125T^{2} \) |
| 7 | \( 1 + 1.77T + 343T^{2} \) |
| 11 | \( 1 + 48.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.22T + 2.19e3T^{2} \) |
| 17 | \( 1 - 34.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 37.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 61.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 164.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 165.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 572.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 543.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 811.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 759.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 973.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 814.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 379.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 87.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 46.6T + 9.12e5T^{2} \) |
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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
−8.464690885247981940448444821860, −7.74108526602057969412094832798, −6.89484005969079053259693210169, −5.77587613240125555579217988504, −5.50886809462923766694683423553, −4.56042867851649985015332656422, −3.11902011848613314061275842880, −2.60493701803437284423164880060, −1.19418256649454417993238471225, 0,
1.19418256649454417993238471225, 2.60493701803437284423164880060, 3.11902011848613314061275842880, 4.56042867851649985015332656422, 5.50886809462923766694683423553, 5.77587613240125555579217988504, 6.89484005969079053259693210169, 7.74108526602057969412094832798, 8.464690885247981940448444821860
