| L(s) = 1 | + 3.04·2-s + 1.42·3-s + 1.28·4-s − 10.9·5-s + 4.35·6-s − 8.62·7-s − 20.4·8-s − 24.9·9-s − 33.3·10-s + 42.2·11-s + 1.84·12-s + 78.4·13-s − 26.2·14-s − 15.6·15-s − 72.6·16-s + 75.8·17-s − 76.0·18-s + 107.·19-s − 14.1·20-s − 12.3·21-s + 128.·22-s − 27.7·23-s − 29.2·24-s − 5.15·25-s + 239.·26-s − 74.2·27-s − 11.1·28-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.275·3-s + 0.161·4-s − 0.979·5-s + 0.296·6-s − 0.465·7-s − 0.903·8-s − 0.924·9-s − 1.05·10-s + 1.15·11-s + 0.0443·12-s + 1.67·13-s − 0.501·14-s − 0.269·15-s − 1.13·16-s + 1.08·17-s − 0.995·18-s + 1.30·19-s − 0.157·20-s − 0.128·21-s + 1.24·22-s − 0.251·23-s − 0.248·24-s − 0.0412·25-s + 1.80·26-s − 0.529·27-s − 0.0749·28-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 | \( 1 - 3.04T + 8T^{2} \) |
| 3 | \( 1 - 1.42T + 27T^{2} \) |
| 5 | \( 1 + 10.9T + 125T^{2} \) |
| 7 | \( 1 + 8.62T + 343T^{2} \) |
| 11 | \( 1 - 42.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 12.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 215.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 598.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 368.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 62.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 186.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 717.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 474.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 632.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 6.66T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.02e3T + 9.12e5T^{2} \) |
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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
−8.506900851693786457870666797947, −7.75306398969585870074586216126, −6.64922942284120993287230044702, −5.93916948060383000142166356625, −5.26915309853831867053585440247, −4.02281871540690559756304518125, −3.52389334448573070306649435753, −3.10973081149963582967834222517, −1.30545585797618303646114879217, 0,
1.30545585797618303646114879217, 3.10973081149963582967834222517, 3.52389334448573070306649435753, 4.02281871540690559756304518125, 5.26915309853831867053585440247, 5.93916948060383000142166356625, 6.64922942284120993287230044702, 7.75306398969585870074586216126, 8.506900851693786457870666797947
