| L(s) = 1 | + 5.07·2-s − 8.66·3-s + 17.7·4-s − 2.61·5-s − 43.9·6-s − 15.1·7-s + 49.3·8-s + 48.1·9-s − 13.2·10-s + 12.6·11-s − 153.·12-s + 46.9·13-s − 76.6·14-s + 22.6·15-s + 108.·16-s − 28.9·17-s + 244.·18-s − 19·19-s − 46.3·20-s + 130.·21-s + 63.9·22-s − 112.·23-s − 427.·24-s − 118.·25-s + 238.·26-s − 183.·27-s − 267.·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 1.66·3-s + 2.21·4-s − 0.234·5-s − 2.99·6-s − 0.815·7-s + 2.17·8-s + 1.78·9-s − 0.419·10-s + 0.345·11-s − 3.69·12-s + 1.00·13-s − 1.46·14-s + 0.390·15-s + 1.69·16-s − 0.413·17-s + 3.19·18-s − 0.229·19-s − 0.518·20-s + 1.36·21-s + 0.620·22-s − 1.01·23-s − 3.63·24-s − 0.945·25-s + 1.79·26-s − 1.30·27-s − 1.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.611990039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.611990039\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 5.07T + 8T^{2} \) |
| 3 | \( 1 + 8.66T + 27T^{2} \) |
| 5 | \( 1 + 2.61T + 125T^{2} \) |
| 7 | \( 1 + 15.1T + 343T^{2} \) |
| 11 | \( 1 - 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 46.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.9T + 4.91e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 57.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 341.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 327.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 459.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 320.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 9.54T + 3.57e5T^{2} \) |
| 73 | \( 1 - 320.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 89.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 439.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 883.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.70e3T + 9.12e5T^{2} \) |
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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
−17.71753620385985026386291273316, −16.16888560917391758379492595941, −15.72836041583817526130278755856, −13.80453494862963605565654409667, −12.55946505828843771273162685542, −11.75231523569651038041613208500, −10.59158517759838794819515282478, −6.66970108304130382083214591104, −5.80090437546682022466286782719, −4.11894757696298139562956381698,
4.11894757696298139562956381698, 5.80090437546682022466286782719, 6.66970108304130382083214591104, 10.59158517759838794819515282478, 11.75231523569651038041613208500, 12.55946505828843771273162685542, 13.80453494862963605565654409667, 15.72836041583817526130278755856, 16.16888560917391758379492595941, 17.71753620385985026386291273316
