| L(s) = 1 | + 2-s + 2·4-s − 6·5-s + 2·7-s + 5·8-s − 6·10-s + 2·11-s + 2·13-s + 2·14-s + 5·16-s − 6·17-s − 4·19-s − 12·20-s + 2·22-s + 2·23-s + 17·25-s + 2·26-s + 4·28-s − 2·29-s − 2·31-s + 10·32-s − 6·34-s − 12·35-s + 2·37-s − 4·38-s − 30·40-s + 4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 2.68·5-s + 0.755·7-s + 1.76·8-s − 1.89·10-s + 0.603·11-s + 0.554·13-s + 0.534·14-s + 5/4·16-s − 1.45·17-s − 0.917·19-s − 2.68·20-s + 0.426·22-s + 0.417·23-s + 17/5·25-s + 0.392·26-s + 0.755·28-s − 0.371·29-s − 0.359·31-s + 1.76·32-s − 1.02·34-s − 2.02·35-s + 0.328·37-s − 0.648·38-s − 4.74·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.267570575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.267570575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 110 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 194 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 222 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−11.20363639310005632484633214780, −10.66913523976373548693074771689, −9.966496820274017928775391258543, −9.350105952498327959702765301853, −8.540067563817448895495835929655, −8.254905317298261597097947285807, −8.251434300043529018213532833211, −7.53518203195715746265288408686, −7.16575431738473006031599528774, −6.81147182438356980663234845793, −6.51170113575159198997059072050, −5.67153197742313612986465513592, −4.88364377100089667363934438738, −4.62752241095814088830050221313, −4.13592854823371659674728409848, −3.79894712507237977365226115983, −3.45514576744159127624363870826, −2.39181914087941912708702112316, −1.82898168095775629493084590715, −0.72687725945129004235660609447,
0.72687725945129004235660609447, 1.82898168095775629493084590715, 2.39181914087941912708702112316, 3.45514576744159127624363870826, 3.79894712507237977365226115983, 4.13592854823371659674728409848, 4.62752241095814088830050221313, 4.88364377100089667363934438738, 5.67153197742313612986465513592, 6.51170113575159198997059072050, 6.81147182438356980663234845793, 7.16575431738473006031599528774, 7.53518203195715746265288408686, 8.251434300043529018213532833211, 8.254905317298261597097947285807, 8.540067563817448895495835929655, 9.350105952498327959702765301853, 9.966496820274017928775391258543, 10.66913523976373548693074771689, 11.20363639310005632484633214780
