| L(s) = 1 | + 1.52·2-s − 5.66·4-s + 19.3·5-s − 4.84·7-s − 20.8·8-s + 29.5·10-s − 61.0·11-s − 7.39·14-s + 13.5·16-s + 41.7·17-s + 107.·19-s − 109.·20-s − 93.2·22-s − 28.5·23-s + 249.·25-s + 27.4·28-s + 89.8·29-s − 183.·31-s + 187.·32-s + 63.7·34-s − 93.6·35-s − 418.·37-s + 164.·38-s − 403.·40-s − 142.·41-s − 71.0·43-s + 346.·44-s + ⋯ |
| L(s) = 1 | + 0.539·2-s − 0.708·4-s + 1.72·5-s − 0.261·7-s − 0.922·8-s + 0.933·10-s − 1.67·11-s − 0.141·14-s + 0.211·16-s + 0.596·17-s + 1.29·19-s − 1.22·20-s − 0.903·22-s − 0.258·23-s + 1.99·25-s + 0.185·28-s + 0.575·29-s − 1.06·31-s + 1.03·32-s + 0.321·34-s − 0.452·35-s − 1.85·37-s + 0.700·38-s − 1.59·40-s − 0.543·41-s − 0.252·43-s + 1.18·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.52T + 8T^{2} \) |
| 5 | \( 1 - 19.3T + 125T^{2} \) |
| 7 | \( 1 + 4.84T + 343T^{2} \) |
| 11 | \( 1 + 61.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 41.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 71.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 25.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 684.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 672.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 326.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 24.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 108.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−8.892436435165146964166282501710, −7.958816907261189542599942445383, −6.92639283346978393435394052734, −5.78758474154498648997211127016, −5.45251695268988356899321811826, −4.84698830675110083589147961760, −3.36525973481576649496394040510, −2.68097238726124859097285883993, −1.46550902759585952670886832760, 0,
1.46550902759585952670886832760, 2.68097238726124859097285883993, 3.36525973481576649496394040510, 4.84698830675110083589147961760, 5.45251695268988356899321811826, 5.78758474154498648997211127016, 6.92639283346978393435394052734, 7.958816907261189542599942445383, 8.892436435165146964166282501710
