| L(s) = 1 | + 2.25·2-s − 4.02·3-s − 2.93·4-s − 4.98·5-s − 9.06·6-s − 5.25·7-s − 24.6·8-s − 10.7·9-s − 11.2·10-s + 25.9·11-s + 11.8·12-s + 14.2·13-s − 11.8·14-s + 20.0·15-s − 31.9·16-s + 20.3·17-s − 24.2·18-s + 2.21·19-s + 14.6·20-s + 21.1·21-s + 58.4·22-s + 39.1·23-s + 99.1·24-s − 100.·25-s + 31.9·26-s + 152.·27-s + 15.3·28-s + ⋯ |
| L(s) = 1 | + 0.795·2-s − 0.775·3-s − 0.366·4-s − 0.446·5-s − 0.616·6-s − 0.283·7-s − 1.08·8-s − 0.399·9-s − 0.355·10-s + 0.711·11-s + 0.284·12-s + 0.303·13-s − 0.225·14-s + 0.345·15-s − 0.499·16-s + 0.290·17-s − 0.317·18-s + 0.0267·19-s + 0.163·20-s + 0.219·21-s + 0.566·22-s + 0.354·23-s + 0.843·24-s − 0.800·25-s + 0.241·26-s + 1.08·27-s + 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.25T + 8T^{2} \) |
| 3 | \( 1 + 4.02T + 27T^{2} \) |
| 5 | \( 1 + 4.98T + 125T^{2} \) |
| 7 | \( 1 + 5.25T + 343T^{2} \) |
| 11 | \( 1 - 25.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 14.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.21T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 277.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 47.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 43.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 98.7T + 6.89e4T^{2} \) |
| 47 | \( 1 - 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 230.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 400.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 640.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 251.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 526.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 978.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 37.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 437.T + 7.04e5T^{2} \) |
| 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
−8.591164466858738421611342889865, −7.64080146739220386019652847421, −6.44050430492123356409189843610, −6.09172966612277867247623052702, −5.17450000649227853547351914228, −4.44316952290746808682489055716, −3.62276132536920435398048010830, −2.77944483924923183773076615671, −1.01548886004882102303929666389, 0,
1.01548886004882102303929666389, 2.77944483924923183773076615671, 3.62276132536920435398048010830, 4.44316952290746808682489055716, 5.17450000649227853547351914228, 6.09172966612277867247623052702, 6.44050430492123356409189843610, 7.64080146739220386019652847421, 8.591164466858738421611342889865
