| L(s) = 1 | + 3.07·2-s + 5.77·3-s + 1.48·4-s + 13.4·5-s + 17.7·6-s − 34.2·7-s − 20.0·8-s + 6.35·9-s + 41.4·10-s + 24.2·11-s + 8.55·12-s + 11.8·13-s − 105.·14-s + 77.8·15-s − 73.6·16-s + 110.·17-s + 19.5·18-s − 45.6·19-s + 19.9·20-s − 197.·21-s + 74.8·22-s − 214.·23-s − 115.·24-s + 56.5·25-s + 36.3·26-s − 119.·27-s − 50.7·28-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 1.11·3-s + 0.185·4-s + 1.20·5-s + 1.20·6-s − 1.84·7-s − 0.887·8-s + 0.235·9-s + 1.31·10-s + 0.665·11-s + 0.205·12-s + 0.252·13-s − 2.01·14-s + 1.33·15-s − 1.15·16-s + 1.58·17-s + 0.256·18-s − 0.551·19-s + 0.223·20-s − 2.05·21-s + 0.724·22-s − 1.94·23-s − 0.985·24-s + 0.452·25-s + 0.274·26-s − 0.849·27-s − 0.342·28-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 | \( 1 - 3.07T + 8T^{2} \) |
| 3 | \( 1 - 5.77T + 27T^{2} \) |
| 5 | \( 1 - 13.4T + 125T^{2} \) |
| 7 | \( 1 + 34.2T + 343T^{2} \) |
| 11 | \( 1 - 24.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 214.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 57.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 0.775T + 6.89e4T^{2} \) |
| 47 | \( 1 + 146.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 282.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 698.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 98.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 517.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 29.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 75.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 882.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 716.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 142.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 352.T + 9.12e5T^{2} \) |
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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
−8.735457714736029219786493725532, −7.68862719062665194482381837981, −6.38935038881063016620077753388, −6.13925059794967472644081738255, −5.40531507370851435133174313746, −3.95307734801759411056186416329, −3.46605369207622023238838256486, −2.78934564583340686575873725804, −1.79671012116075412216482574670, 0,
1.79671012116075412216482574670, 2.78934564583340686575873725804, 3.46605369207622023238838256486, 3.95307734801759411056186416329, 5.40531507370851435133174313746, 6.13925059794967472644081738255, 6.38935038881063016620077753388, 7.68862719062665194482381837981, 8.735457714736029219786493725532
