| L(s) = 1 | + 5.48·3-s + 7.85·5-s + 7.65·7-s + 3.13·9-s + 2.42·11-s − 62.4·13-s + 43.1·15-s − 117.·17-s + 76.2·19-s + 42.0·21-s − 23·23-s − 63.2·25-s − 131.·27-s − 39.2·29-s − 171.·31-s + 13.3·33-s + 60.1·35-s + 280.·37-s − 342.·39-s − 280.·41-s − 393.·43-s + 24.6·45-s + 467.·47-s − 284.·49-s − 642.·51-s − 253.·53-s + 19.0·55-s + ⋯ |
| L(s) = 1 | + 1.05·3-s + 0.702·5-s + 0.413·7-s + 0.116·9-s + 0.0664·11-s − 1.33·13-s + 0.742·15-s − 1.66·17-s + 0.920·19-s + 0.436·21-s − 0.208·23-s − 0.506·25-s − 0.933·27-s − 0.251·29-s − 0.995·31-s + 0.0702·33-s + 0.290·35-s + 1.24·37-s − 1.40·39-s − 1.06·41-s − 1.39·43-s + 0.0816·45-s + 1.45·47-s − 0.829·49-s − 1.76·51-s − 0.656·53-s + 0.0467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 5.48T + 27T^{2} \) |
| 5 | \( 1 - 7.85T + 125T^{2} \) |
| 7 | \( 1 - 7.65T + 343T^{2} \) |
| 11 | \( 1 - 2.42T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 39.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 171.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 280.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 467.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 253.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 850.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 176.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 684.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 535.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 323.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 327.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−8.864969754372233457840796204721, −7.949974610440991867566653211126, −7.32812802138215114433187846550, −6.33583881540520041636291417471, −5.31951813621714904934739771577, −4.50889868843394077189249043663, −3.35774937188101063063408846775, −2.36628810954434616002290239018, −1.80421188699432600531460994462, 0,
1.80421188699432600531460994462, 2.36628810954434616002290239018, 3.35774937188101063063408846775, 4.50889868843394077189249043663, 5.31951813621714904934739771577, 6.33583881540520041636291417471, 7.32812802138215114433187846550, 7.949974610440991867566653211126, 8.864969754372233457840796204721
