| L(s) = 1 | + 4.71·2-s − 3·3-s + 14.2·4-s + 1.63·5-s − 14.1·6-s − 2.96·7-s + 29.3·8-s + 9·9-s + 7.70·10-s + 11·11-s − 42.6·12-s − 70.7·13-s − 13.9·14-s − 4.90·15-s + 24.4·16-s + 63.7·17-s + 42.4·18-s + 100.·19-s + 23.2·20-s + 8.90·21-s + 51.8·22-s − 196.·23-s − 87.9·24-s − 122.·25-s − 333.·26-s − 27·27-s − 42.2·28-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 0.577·3-s + 1.77·4-s + 0.146·5-s − 0.962·6-s − 0.160·7-s + 1.29·8-s + 0.333·9-s + 0.243·10-s + 0.301·11-s − 1.02·12-s − 1.50·13-s − 0.267·14-s − 0.0843·15-s + 0.382·16-s + 0.909·17-s + 0.555·18-s + 1.21·19-s + 0.259·20-s + 0.0925·21-s + 0.502·22-s − 1.78·23-s − 0.748·24-s − 0.978·25-s − 2.51·26-s − 0.192·27-s − 0.285·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 4.71T + 8T^{2} \) |
| 5 | \( 1 - 1.63T + 125T^{2} \) |
| 7 | \( 1 + 2.96T + 343T^{2} \) |
| 13 | \( 1 + 70.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 196.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 282.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 10.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 83.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 542.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 363.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 328.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 103.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 511.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.05T + 9.12e5T^{2} \) |
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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
−8.003081469348797632068672678861, −7.34192054164900728031368702148, −6.49313534114223797805038096394, −5.78157067233213244903056086414, −5.19602686710516605135057423389, −4.43580699316560715431262158120, −3.58499281912608183906867635492, −2.67847533495932595221133946080, −1.62285497288502960132076519992, 0,
1.62285497288502960132076519992, 2.67847533495932595221133946080, 3.58499281912608183906867635492, 4.43580699316560715431262158120, 5.19602686710516605135057423389, 5.78157067233213244903056086414, 6.49313534114223797805038096394, 7.34192054164900728031368702148, 8.003081469348797632068672678861
