| L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s + 2·9-s − 4·11-s − 6·13-s + 4·15-s + 2·19-s + 4·21-s − 8·23-s − 2·25-s − 6·27-s − 4·31-s + 8·33-s + 4·35-s + 12·39-s − 8·41-s + 4·43-s − 4·45-s − 12·47-s + 3·49-s + 20·53-s + 8·55-s − 4·57-s − 14·59-s − 18·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s + 2/3·9-s − 1.20·11-s − 1.66·13-s + 1.03·15-s + 0.458·19-s + 0.872·21-s − 1.66·23-s − 2/5·25-s − 1.15·27-s − 0.718·31-s + 1.39·33-s + 0.676·35-s + 1.92·39-s − 1.24·41-s + 0.609·43-s − 0.596·45-s − 1.75·47-s + 3/7·49-s + 2.74·53-s + 1.07·55-s − 0.529·57-s − 1.82·59-s − 2.30·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.72313516179983614681680939696, −10.59594954393709230171084389256, −9.873065737791384059978413168053, −9.820905988363453519955648919744, −9.295382764232834230960039290466, −8.443774772545977957850657392897, −8.021955820128698147095384107536, −7.43538390169810641438483093375, −7.36064089202582008214152259807, −6.73471311309608163493922866668, −5.85304500617735930094237983656, −5.84517032387388497128965377763, −5.11280686088676126389122607065, −4.67292011763688602369820174270, −4.02873041392259911358913600301, −3.42401183284473340644174108111, −2.67244546277557013275540306472, −1.87684936313832388702670570061, 0, 0,
1.87684936313832388702670570061, 2.67244546277557013275540306472, 3.42401183284473340644174108111, 4.02873041392259911358913600301, 4.67292011763688602369820174270, 5.11280686088676126389122607065, 5.84517032387388497128965377763, 5.85304500617735930094237983656, 6.73471311309608163493922866668, 7.36064089202582008214152259807, 7.43538390169810641438483093375, 8.021955820128698147095384107536, 8.443774772545977957850657392897, 9.295382764232834230960039290466, 9.820905988363453519955648919744, 9.873065737791384059978413168053, 10.59594954393709230171084389256, 10.72313516179983614681680939696
