| L(s) = 1 | − 0.438·2-s − 26.3·3-s − 31.8·4-s + 61.4·5-s + 11.5·6-s − 162.·7-s + 27.9·8-s + 452.·9-s − 26.9·10-s − 361.·11-s + 838.·12-s − 169·13-s + 71.1·14-s − 1.62e3·15-s + 1.00e3·16-s − 1.57e3·17-s − 198.·18-s − 98.2·19-s − 1.95e3·20-s + 4.27e3·21-s + 158.·22-s + 1.60e3·23-s − 737.·24-s + 652.·25-s + 74.0·26-s − 5.52e3·27-s + 5.16e3·28-s + ⋯ |
| L(s) = 1 | − 0.0775·2-s − 1.69·3-s − 0.993·4-s + 1.09·5-s + 0.131·6-s − 1.25·7-s + 0.154·8-s + 1.86·9-s − 0.0852·10-s − 0.899·11-s + 1.68·12-s − 0.277·13-s + 0.0970·14-s − 1.85·15-s + 0.982·16-s − 1.32·17-s − 0.144·18-s − 0.0624·19-s − 1.09·20-s + 2.11·21-s + 0.0697·22-s + 0.633·23-s − 0.261·24-s + 0.208·25-s + 0.0214·26-s − 1.45·27-s + 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 169T \) |
| good | 2 | \( 1 + 0.438T + 32T^{2} \) |
| 3 | \( 1 + 26.3T + 243T^{2} \) |
| 5 | \( 1 - 61.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 162.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 361.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.57e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 98.2T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 307.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 104.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.59e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.01e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.57e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 912.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−17.82840275201267415546276683894, −17.22059136774406444762166152433, −15.88826246171006326113000711110, −13.44174653581734276256283858885, −12.65322547324310671515351252260, −10.58559667747287484577722066462, −9.523134886135161996617041862672, −6.42362539238542465342308265688, −5.09752034417133274464820502541, 0,
5.09752034417133274464820502541, 6.42362539238542465342308265688, 9.523134886135161996617041862672, 10.58559667747287484577722066462, 12.65322547324310671515351252260, 13.44174653581734276256283858885, 15.88826246171006326113000711110, 17.22059136774406444762166152433, 17.82840275201267415546276683894
