| L(s) = 1 | − 2·2-s − 24.9·3-s − 28·4-s − 74.9·5-s + 49.9·6-s + 120·8-s + 381·9-s + 149.·10-s − 284·11-s + 699.·12-s − 524.·13-s + 1.87e3·15-s + 656·16-s + 149.·17-s − 762·18-s − 2.17e3·19-s + 2.09e3·20-s + 568·22-s + 1.49e3·23-s − 2.99e3·24-s + 2.49e3·25-s + 1.04e3·26-s − 3.44e3·27-s − 4.36e3·29-s − 3.74e3·30-s + 6.44e3·31-s − 5.15e3·32-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.60·3-s − 0.875·4-s − 1.34·5-s + 0.566·6-s + 0.662·8-s + 1.56·9-s + 0.473·10-s − 0.707·11-s + 1.40·12-s − 0.860·13-s + 2.14·15-s + 0.640·16-s + 0.125·17-s − 0.554·18-s − 1.38·19-s + 1.17·20-s + 0.250·22-s + 0.589·23-s − 1.06·24-s + 0.797·25-s + 0.304·26-s − 0.910·27-s − 0.964·29-s − 0.759·30-s + 1.20·31-s − 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1878649763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1878649763\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 2T + 32T^{2} \) |
| 3 | \( 1 + 24.9T + 243T^{2} \) |
| 5 | \( 1 + 74.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 284T + 1.61e5T^{2} \) |
| 13 | \( 1 + 524.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 149.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.64e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 524.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.83e5T + 8.58e9T^{2} \) |
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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
−14.87666734531662154492780887845, −13.00774088918867840392777419336, −12.16452426120625643001517762712, −11.07654606259343191825190591051, −10.11000278058710878791113209938, −8.350377717238574837784015840520, −7.10458692726977830579516010790, −5.27889276202149891284105192365, −4.23966316132992123864667505701, −0.40630917411279454399531091558,
0.40630917411279454399531091558, 4.23966316132992123864667505701, 5.27889276202149891284105192365, 7.10458692726977830579516010790, 8.350377717238574837784015840520, 10.11000278058710878791113209938, 11.07654606259343191825190591051, 12.16452426120625643001517762712, 13.00774088918867840392777419336, 14.87666734531662154492780887845
