| L(s) = 1 | + 6.29·3-s + 5.83·5-s − 20.2·7-s + 12.5·9-s + 59.1·11-s − 21.2·13-s + 36.7·15-s − 93.2·17-s + 69.4·19-s − 127.·21-s − 34.2·23-s − 90.8·25-s − 90.7·27-s − 29·29-s − 155.·31-s + 372.·33-s − 118.·35-s − 117.·37-s − 133.·39-s − 325.·41-s − 358.·43-s + 73.4·45-s + 287.·47-s + 67.5·49-s − 586.·51-s + 496.·53-s + 345.·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 0.522·5-s − 1.09·7-s + 0.465·9-s + 1.62·11-s − 0.453·13-s + 0.632·15-s − 1.33·17-s + 0.838·19-s − 1.32·21-s − 0.310·23-s − 0.727·25-s − 0.646·27-s − 0.185·29-s − 0.898·31-s + 1.96·33-s − 0.571·35-s − 0.521·37-s − 0.549·39-s − 1.24·41-s − 1.27·43-s + 0.243·45-s + 0.892·47-s + 0.196·49-s − 1.61·51-s + 1.28·53-s + 0.847·55-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 | \( 1 - 6.29T + 27T^{2} \) |
| 5 | \( 1 - 5.83T + 125T^{2} \) |
| 7 | \( 1 + 20.2T + 343T^{2} \) |
| 11 | \( 1 - 59.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 34.2T + 1.21e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 358.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 287.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 496.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 474.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 136.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 42.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 266.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 962.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 709.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 413.T + 9.12e5T^{2} \) |
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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
−8.782855444690171854017981432234, −7.75689715851637566026366610763, −6.82188876342972768188019552604, −6.35969162349904090961182456758, −5.26190660410232623467982398960, −3.93227634635400021816749345084, −3.47308319686156569495406977651, −2.42069375554332488388597857296, −1.61747383993137737663772829777, 0,
1.61747383993137737663772829777, 2.42069375554332488388597857296, 3.47308319686156569495406977651, 3.93227634635400021816749345084, 5.26190660410232623467982398960, 6.35969162349904090961182456758, 6.82188876342972768188019552604, 7.75689715851637566026366610763, 8.782855444690171854017981432234
