| L(s) = 1 | − 36·3-s − 1.96e5·5-s − 3.59e7·7-s − 1.16e9·9-s − 1.20e10·11-s − 4.55e10·13-s + 7.06e6·15-s + 4.96e11·17-s + 1.41e12·19-s + 1.29e9·21-s − 7.03e12·23-s − 1.90e13·25-s + 8.36e10·27-s + 3.89e13·29-s + 1.73e14·31-s + 4.32e11·33-s + 7.04e12·35-s − 1.10e15·37-s + 1.63e12·39-s − 1.44e15·41-s + 4.64e15·43-s + 2.28e14·45-s + 8.95e15·47-s − 1.01e16·49-s − 1.78e13·51-s − 3.29e16·53-s + 2.35e15·55-s + ⋯ |
| L(s) = 1 | − 0.00105·3-s − 0.0449·5-s − 0.336·7-s − 0.999·9-s − 1.53·11-s − 1.19·13-s + 4.74e−5·15-s + 1.01·17-s + 1.00·19-s + 0.000355·21-s − 0.814·23-s − 0.997·25-s + 0.00211·27-s + 0.499·29-s + 1.17·31-s + 0.00162·33-s + 0.0151·35-s − 1.40·37-s + 0.00125·39-s − 0.689·41-s + 1.40·43-s + 0.0449·45-s + 1.16·47-s − 0.886·49-s − 0.00107·51-s − 1.37·53-s + 0.0690·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 4 p^{2} T + p^{19} T^{2} \) |
| 5 | \( 1 + 39258 p T + p^{19} T^{2} \) |
| 7 | \( 1 + 5129368 p T + p^{19} T^{2} \) |
| 11 | \( 1 + 12016099980 T + p^{19} T^{2} \) |
| 13 | \( 1 + 3502281298 p T + p^{19} T^{2} \) |
| 17 | \( 1 - 29209602834 p T + p^{19} T^{2} \) |
| 19 | \( 1 - 1410273986444 T + p^{19} T^{2} \) |
| 23 | \( 1 + 7039745388792 T + p^{19} T^{2} \) |
| 29 | \( 1 - 38996890912134 T + p^{19} T^{2} \) |
| 31 | \( 1 - 173641323230816 T + p^{19} T^{2} \) |
| 37 | \( 1 + 1108106825662306 T + p^{19} T^{2} \) |
| 41 | \( 1 + 1444509198124614 T + p^{19} T^{2} \) |
| 43 | \( 1 - 4646075748354260 T + p^{19} T^{2} \) |
| 47 | \( 1 - 8950457686524048 T + p^{19} T^{2} \) |
| 53 | \( 1 + 32948524384463538 T + p^{19} T^{2} \) |
| 59 | \( 1 - 36999205673523588 T + p^{19} T^{2} \) |
| 61 | \( 1 - 82929105285760742 T + p^{19} T^{2} \) |
| 67 | \( 1 + 186668590860047716 T + p^{19} T^{2} \) |
| 71 | \( 1 + 596514630027659112 T + p^{19} T^{2} \) |
| 73 | \( 1 - 310786775495585306 T + p^{19} T^{2} \) |
| 79 | \( 1 - 700397513485701872 T + p^{19} T^{2} \) |
| 83 | \( 1 + 1357882121724855732 T + p^{19} T^{2} \) |
| 89 | \( 1 + 5991411253779123894 T + p^{19} T^{2} \) |
| 97 | \( 1 - 4531118407744664354 T + p^{19} T^{2} \) |
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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
−19.26941935981035577670017541251, −17.53880090469127769168082699389, −15.84302603185983963206918036044, −14.00242865976593738268649036394, −12.09377577414268594318206158221, −10.04266228236597948181254349650, −7.83771734909093384451686067218, −5.43169587716586273269451001365, −2.80218703244253888858692504687, 0,
2.80218703244253888858692504687, 5.43169587716586273269451001365, 7.83771734909093384451686067218, 10.04266228236597948181254349650, 12.09377577414268594318206158221, 14.00242865976593738268649036394, 15.84302603185983963206918036044, 17.53880090469127769168082699389, 19.26941935981035577670017541251
