| L(s) = 1 | − 5·2-s − 28·3-s − 41·4-s − 42·5-s + 140·6-s − 36·7-s + 375·8-s + 255·9-s + 210·10-s − 376·11-s + 1.14e3·12-s − 338·13-s + 180·14-s + 1.17e3·15-s + 607·16-s − 2.63e3·17-s − 1.27e3·18-s − 312·19-s + 1.72e3·20-s + 1.00e3·21-s + 1.88e3·22-s − 2.62e3·23-s − 1.05e4·24-s + 1.87e3·25-s + 1.69e3·26-s + 868·27-s + 1.47e3·28-s + ⋯ |
| L(s) = 1 | − 0.883·2-s − 1.79·3-s − 1.28·4-s − 0.751·5-s + 1.58·6-s − 0.277·7-s + 2.07·8-s + 1.04·9-s + 0.664·10-s − 0.936·11-s + 2.30·12-s − 0.554·13-s + 0.245·14-s + 1.34·15-s + 0.592·16-s − 2.20·17-s − 0.927·18-s − 0.198·19-s + 0.962·20-s + 0.498·21-s + 0.828·22-s − 1.03·23-s − 3.72·24-s + 0.599·25-s + 0.490·26-s + 0.229·27-s + 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 5 T + 33 p T^{2} + 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 28 T + 529 T^{2} + 28 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 42 T - 109 T^{2} + 42 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 36 T + 13113 T^{2} + 36 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 376 T + 327458 T^{2} + 376 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2630 T + 4499307 T^{2} + 2630 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 312 T + 4973202 T^{2} + 312 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2624 T + 6072542 T^{2} + 2624 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 28 p T + 41177342 T^{2} + 28 p^{6} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7720 T + 71304910 T^{2} - 7720 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 16858 T + 195347155 T^{2} + 16858 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7840 T + 230885010 T^{2} - 7840 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2420 T + 200878009 T^{2} - 2420 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9972 T + 86137385 T^{2} - 9972 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 43720 T + 1116804698 T^{2} + 43720 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 38936 T + 1489531170 T^{2} + 38936 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1984 T + 1322910298 T^{2} - 1984 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 69928 T + 3922568242 T^{2} + 69928 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 67396 T + 4686895793 T^{2} - 67396 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 74412 T + 4066416822 T^{2} - 74412 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 55296 T + 6515668494 T^{2} + 55296 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 75712 T + 7946370822 T^{2} + 75712 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 116508 T + 12213785846 T^{2} - 116508 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 34756 T + 2822512198 T^{2} + 34756 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.21641402743395835008708639836, −17.82840275201267415546276683894, −17.22059136774406444762166152433, −17.04546383003913073150984208649, −15.88826246171006326113000711110, −15.62452668382559574219879508070, −14.19373837112560839413522349369, −13.44174653581734276256283858885, −12.65322547324310671515351252260, −11.96857811561027895312429109720, −10.89265740864086131542400088784, −10.58559667747287484577722066462, −9.523134886135161996617041862672, −8.604094016517240757911341303022, −7.76406369095647844737940227415, −6.42362539238542465342308265688, −5.09752034417133274464820502541, −4.42812287454429366295353399292, 0, 0,
4.42812287454429366295353399292, 5.09752034417133274464820502541, 6.42362539238542465342308265688, 7.76406369095647844737940227415, 8.604094016517240757911341303022, 9.523134886135161996617041862672, 10.58559667747287484577722066462, 10.89265740864086131542400088784, 11.96857811561027895312429109720, 12.65322547324310671515351252260, 13.44174653581734276256283858885, 14.19373837112560839413522349369, 15.62452668382559574219879508070, 15.88826246171006326113000711110, 17.04546383003913073150984208649, 17.22059136774406444762166152433, 17.82840275201267415546276683894, 18.21641402743395835008708639836
