| L(s) = 1 | + 2·2-s + 5·5-s − 8·7-s − 8·8-s + 10·10-s + 18·11-s − 8·13-s − 16·14-s − 16·16-s − 30·17-s + 46·19-s + 36·22-s + 63·23-s − 16·26-s + 156·29-s + 85·31-s − 60·34-s − 40·35-s + 148·37-s + 92·38-s − 40·40-s + 246·41-s + 190·43-s + 126·46-s + 288·47-s + 343·49-s + 354·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.447·5-s − 0.431·7-s − 0.353·8-s + 0.316·10-s + 0.493·11-s − 0.170·13-s − 0.305·14-s − 1/4·16-s − 0.428·17-s + 0.555·19-s + 0.348·22-s + 0.571·23-s − 0.120·26-s + 0.998·29-s + 0.492·31-s − 0.302·34-s − 0.193·35-s + 0.657·37-s + 0.392·38-s − 0.158·40-s + 0.937·41-s + 0.673·43-s + 0.403·46-s + 0.893·47-s + 49-s + 0.917·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.830164064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.830164064\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 8 T - 279 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T - 1007 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T - 2133 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 23 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 63 T - 8198 T^{2} - 63 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 156 T - 53 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 85 T - 22566 T^{2} - 85 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 p T - 5 p^{2} T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 190 T - 43407 T^{2} - 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 288 T - 20879 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 177 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 792 T + 421885 T^{2} - 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 907 T + 595668 T^{2} - 907 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 322 T - 197079 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 270 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 254 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 1123 T + 768090 T^{2} - 1123 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 771 T + 22654 T^{2} + 771 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 198 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1192 T + 508191 T^{2} - 1192 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.853994932652663345573391171505, −9.791208665020607016791965622670, −9.130727136482495508239635445896, −9.046197864118827264268066766883, −8.195499868767859103997151114182, −8.192364674537972404821284297947, −7.20534239157437740001057830233, −7.03174231313873290511570985489, −6.54805288700156006673350911120, −6.10165830095966074734235498354, −5.51778033297934997925734947212, −5.33419810598634331793638516314, −4.62989078298390792494034139559, −4.16026630542037856290070345648, −3.76391114935019560272245594015, −3.13463225649381368953701464888, −2.41291032873386159073496627825, −2.24518839457284359349751334556, −0.838656169804247970741539868723, −0.825812935410452080035683721492,
0.825812935410452080035683721492, 0.838656169804247970741539868723, 2.24518839457284359349751334556, 2.41291032873386159073496627825, 3.13463225649381368953701464888, 3.76391114935019560272245594015, 4.16026630542037856290070345648, 4.62989078298390792494034139559, 5.33419810598634331793638516314, 5.51778033297934997925734947212, 6.10165830095966074734235498354, 6.54805288700156006673350911120, 7.03174231313873290511570985489, 7.20534239157437740001057830233, 8.192364674537972404821284297947, 8.195499868767859103997151114182, 9.046197864118827264268066766883, 9.130727136482495508239635445896, 9.791208665020607016791965622670, 9.853994932652663345573391171505
