| L(s) = 1 | − 5.42·3-s − 1.04·5-s − 25.1·7-s + 2.41·9-s + 25.3·11-s + 84.9·13-s + 5.67·15-s + 84.8·17-s + 0.837·19-s + 136.·21-s − 62.2·23-s − 123.·25-s + 133.·27-s − 167.·29-s + 191.·31-s − 137.·33-s + 26.2·35-s − 37·37-s − 460.·39-s − 223.·41-s − 314.·43-s − 2.53·45-s + 357.·47-s + 287.·49-s − 460.·51-s − 123.·53-s − 26.5·55-s + ⋯ |
| L(s) = 1 | − 1.04·3-s − 0.0936·5-s − 1.35·7-s + 0.0895·9-s + 0.693·11-s + 1.81·13-s + 0.0977·15-s + 1.21·17-s + 0.0101·19-s + 1.41·21-s − 0.563·23-s − 0.991·25-s + 0.950·27-s − 1.07·29-s + 1.11·31-s − 0.724·33-s + 0.126·35-s − 0.164·37-s − 1.89·39-s − 0.849·41-s − 1.11·43-s − 0.00838·45-s + 1.10·47-s + 0.836·49-s − 1.26·51-s − 0.320·53-s − 0.0649·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + 37T \) |
| good | 3 | \( 1 + 5.42T + 27T^{2} \) |
| 5 | \( 1 + 1.04T + 125T^{2} \) |
| 7 | \( 1 + 25.1T + 343T^{2} \) |
| 11 | \( 1 - 25.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 0.837T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 191.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 223.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 314.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 333.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 60.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 262.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 957.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 169.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 2.57T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−9.997507297098504006243759284260, −9.084313675156436776201038677369, −8.093264559495826701760067739399, −6.78319067512922749351045485032, −6.10442927366105767397386955259, −5.59347188960315785328300099769, −3.99766754743861611167012996921, −3.23661420569627006391981699034, −1.26394079589048931686062621577, 0,
1.26394079589048931686062621577, 3.23661420569627006391981699034, 3.99766754743861611167012996921, 5.59347188960315785328300099769, 6.10442927366105767397386955259, 6.78319067512922749351045485032, 8.093264559495826701760067739399, 9.084313675156436776201038677369, 9.997507297098504006243759284260
