| L(s) = 1 | + 729·3-s + 3.02e4·5-s + 2.35e5·7-s + 5.31e5·9-s + 1.11e7·11-s − 8.04e6·13-s + 2.20e7·15-s − 1.17e8·17-s + 2.14e8·19-s + 1.71e8·21-s + 8.30e8·23-s − 3.08e8·25-s + 3.87e8·27-s + 1.25e9·29-s + 6.15e9·31-s + 8.15e9·33-s + 7.10e9·35-s + 5.49e9·37-s − 5.86e9·39-s − 4.67e9·41-s − 7.11e9·43-s + 1.60e10·45-s − 2.95e10·47-s − 4.16e10·49-s − 8.56e10·51-s + 2.04e11·53-s + 3.37e11·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.864·5-s + 0.755·7-s + 1/3·9-s + 1.90·11-s − 0.462·13-s + 0.499·15-s − 1.18·17-s + 1.04·19-s + 0.436·21-s + 1.16·23-s − 0.252·25-s + 0.192·27-s + 0.390·29-s + 1.24·31-s + 1.09·33-s + 0.653·35-s + 0.352·37-s − 0.267·39-s − 0.153·41-s − 0.171·43-s + 0.288·45-s − 0.399·47-s − 0.429·49-s − 0.681·51-s + 1.26·53-s + 1.64·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(5.024307436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.024307436\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{6} T \) |
| good | 5 | \( 1 - 6042 p T + p^{13} T^{2} \) |
| 7 | \( 1 - 33584 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 1016628 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 8049614 T + p^{13} T^{2} \) |
| 17 | \( 1 + 117494622 T + p^{13} T^{2} \) |
| 19 | \( 1 - 214061380 T + p^{13} T^{2} \) |
| 23 | \( 1 - 830555544 T + p^{13} T^{2} \) |
| 29 | \( 1 - 1252400250 T + p^{13} T^{2} \) |
| 31 | \( 1 - 6159350552 T + p^{13} T^{2} \) |
| 37 | \( 1 - 5498191402 T + p^{13} T^{2} \) |
| 41 | \( 1 + 4678687878 T + p^{13} T^{2} \) |
| 43 | \( 1 + 7115013764 T + p^{13} T^{2} \) |
| 47 | \( 1 + 29528776992 T + p^{13} T^{2} \) |
| 53 | \( 1 - 204125042466 T + p^{13} T^{2} \) |
| 59 | \( 1 - 29909821020 T + p^{13} T^{2} \) |
| 61 | \( 1 - 134392006738 T + p^{13} T^{2} \) |
| 67 | \( 1 + 348518801948 T + p^{13} T^{2} \) |
| 71 | \( 1 - 1314335409192 T + p^{13} T^{2} \) |
| 73 | \( 1 + 1178875922326 T + p^{13} T^{2} \) |
| 79 | \( 1 + 1072420659640 T + p^{13} T^{2} \) |
| 83 | \( 1 + 1124025139644 T + p^{13} T^{2} \) |
| 89 | \( 1 - 2235610909530 T + p^{13} T^{2} \) |
| 97 | \( 1 + 14215257165502 T + p^{13} T^{2} \) |
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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
−9.931705645204870144344295543941, −9.234705201872295630936672225663, −8.460624915426960037412976781928, −7.14365290473944814198097572481, −6.34199932816638361849989122989, −5.01843729101194198546732365451, −4.08029806067487384664622141437, −2.77567449513918282308454166358, −1.72909153514084774949723204262, −0.995451122825397055342904084106,
0.995451122825397055342904084106, 1.72909153514084774949723204262, 2.77567449513918282308454166358, 4.08029806067487384664622141437, 5.01843729101194198546732365451, 6.34199932816638361849989122989, 7.14365290473944814198097572481, 8.460624915426960037412976781928, 9.234705201872295630936672225663, 9.931705645204870144344295543941
