| L(s) = 1 | + 2-s − 3-s + 2·5-s − 6-s − 5·7-s − 8-s + 2·10-s − 3·11-s − 2·13-s − 5·14-s − 2·15-s − 16-s − 5·17-s − 19-s + 5·21-s − 3·22-s + 24-s + 5·25-s − 2·26-s + 27-s − 2·29-s − 2·30-s − 4·31-s + 3·33-s − 5·34-s − 10·35-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.894·5-s − 0.408·6-s − 1.88·7-s − 0.353·8-s + 0.632·10-s − 0.904·11-s − 0.554·13-s − 1.33·14-s − 0.516·15-s − 1/4·16-s − 1.21·17-s − 0.229·19-s + 1.09·21-s − 0.639·22-s + 0.204·24-s + 25-s − 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.365·30-s − 0.718·31-s + 0.522·33-s − 0.857·34-s − 1.69·35-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9595790072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9595790072\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−11.03779459159792050806087919746, −10.44207720444818136592454567071, −10.28313845173251812869786769390, −9.686186313212732169722741890781, −9.406343746102047802525323498436, −8.908850760181139815147167552718, −8.622139728649890942531257274479, −7.62525600952476278023484431714, −7.34544332234098272794883785145, −6.62312608806841709158313106572, −6.40540042738356625626577830248, −5.90676224648613412199363688981, −5.63331278480893947610050772058, −4.77815757539992127924390644440, −4.73291315000348340617855276648, −3.71487970712276797995271084164, −3.28162877485795102032516129313, −2.52880618990723688704929060118, −2.16562012565718001178786997235, −0.49527662980868924351165593227,
0.49527662980868924351165593227, 2.16562012565718001178786997235, 2.52880618990723688704929060118, 3.28162877485795102032516129313, 3.71487970712276797995271084164, 4.73291315000348340617855276648, 4.77815757539992127924390644440, 5.63331278480893947610050772058, 5.90676224648613412199363688981, 6.40540042738356625626577830248, 6.62312608806841709158313106572, 7.34544332234098272794883785145, 7.62525600952476278023484431714, 8.622139728649890942531257274479, 8.908850760181139815147167552718, 9.406343746102047802525323498436, 9.686186313212732169722741890781, 10.28313845173251812869786769390, 10.44207720444818136592454567071, 11.03779459159792050806087919746
