| L(s) = 1 | + 4.75·2-s − 7.02·3-s + 14.5·4-s − 4.22·5-s − 33.3·6-s − 11.1·7-s + 31.2·8-s + 22.2·9-s − 20.0·10-s − 43.0·11-s − 102.·12-s + 62.9·13-s − 53.1·14-s + 29.6·15-s + 31.9·16-s + 136.·17-s + 105.·18-s + 108.·19-s − 61.5·20-s + 78.4·21-s − 204.·22-s − 142.·23-s − 219.·24-s − 107.·25-s + 299.·26-s + 33.0·27-s − 162.·28-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 1.35·3-s + 1.82·4-s − 0.377·5-s − 2.26·6-s − 0.603·7-s + 1.38·8-s + 0.825·9-s − 0.634·10-s − 1.18·11-s − 2.46·12-s + 1.34·13-s − 1.01·14-s + 0.510·15-s + 0.498·16-s + 1.94·17-s + 1.38·18-s + 1.30·19-s − 0.688·20-s + 0.815·21-s − 1.98·22-s − 1.29·23-s − 1.86·24-s − 0.857·25-s + 2.25·26-s + 0.235·27-s − 1.09·28-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 | \( 1 - 4.75T + 8T^{2} \) |
| 3 | \( 1 + 7.02T + 27T^{2} \) |
| 5 | \( 1 + 4.22T + 125T^{2} \) |
| 7 | \( 1 + 11.1T + 343T^{2} \) |
| 11 | \( 1 + 43.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 29.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 69.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 128.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 95.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 416.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 644.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 228.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 877.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 258.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 396.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 916.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 66.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 365.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−8.037535320671109162501593550549, −7.46884096659371791892972957060, −6.25348539919099908966122649761, −5.94360045662655197005588265343, −5.35944034843413985641227238486, −4.55038967471476878015227539131, −3.52029499714626119087668187501, −2.98568322640789666892204351944, −1.31294116059895365460984436392, 0,
1.31294116059895365460984436392, 2.98568322640789666892204351944, 3.52029499714626119087668187501, 4.55038967471476878015227539131, 5.35944034843413985641227238486, 5.94360045662655197005588265343, 6.25348539919099908966122649761, 7.46884096659371791892972957060, 8.037535320671109162501593550549
