| L(s) = 1 | − 4.62·2-s + 0.223·3-s + 13.3·4-s − 5.07·5-s − 1.03·6-s + 13.5·7-s − 24.8·8-s − 26.9·9-s + 23.4·10-s − 11·11-s + 2.98·12-s − 62.6·14-s − 1.13·15-s + 7.77·16-s + 65.3·17-s + 124.·18-s + 25.7·19-s − 67.8·20-s + 3.03·21-s + 50.8·22-s − 99.9·23-s − 5.55·24-s − 99.2·25-s − 12.0·27-s + 181.·28-s + 92.8·29-s + 5.24·30-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 0.0430·3-s + 1.67·4-s − 0.454·5-s − 0.0703·6-s + 0.732·7-s − 1.09·8-s − 0.998·9-s + 0.742·10-s − 0.301·11-s + 0.0719·12-s − 1.19·14-s − 0.0195·15-s + 0.121·16-s + 0.932·17-s + 1.63·18-s + 0.310·19-s − 0.759·20-s + 0.0315·21-s + 0.492·22-s − 0.905·23-s − 0.0472·24-s − 0.793·25-s − 0.0859·27-s + 1.22·28-s + 0.594·29-s + 0.0319·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.62T + 8T^{2} \) |
| 3 | \( 1 - 0.223T + 27T^{2} \) |
| 5 | \( 1 + 5.07T + 125T^{2} \) |
| 7 | \( 1 - 13.5T + 343T^{2} \) |
| 17 | \( 1 - 65.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 99.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 92.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 46.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 51.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 443.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 242.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 104.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 230.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 423.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 556.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 261.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 160.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 906.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 636.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 229.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
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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
−8.308695369004576051230361418880, −7.970498954518307521134920896520, −7.43591564159522731312689291682, −6.27571063723358729740134545806, −5.46900376659710281946511547148, −4.30968255882527992626262060047, −3.05073825416771019770402861376, −2.08885510473540562606695220523, −0.996783995447412952991118154755, 0,
0.996783995447412952991118154755, 2.08885510473540562606695220523, 3.05073825416771019770402861376, 4.30968255882527992626262060047, 5.46900376659710281946511547148, 6.27571063723358729740134545806, 7.43591564159522731312689291682, 7.970498954518307521134920896520, 8.308695369004576051230361418880
