L(s) = 1 | + 2-s + 2·3-s − 2·5-s + 2·6-s − 2·7-s − 8-s − 3·9-s − 2·10-s − 6·11-s − 2·13-s − 2·14-s − 4·15-s − 16-s − 6·17-s − 3·18-s + 4·19-s − 4·21-s − 6·22-s + 6·23-s − 2·24-s + 3·25-s − 2·26-s − 14·27-s + 6·29-s − 4·30-s − 5·31-s − 12·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s − 0.894·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s − 9-s − 0.632·10-s − 1.80·11-s − 0.554·13-s − 0.534·14-s − 1.03·15-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.872·21-s − 1.27·22-s + 1.25·23-s − 0.408·24-s + 3/5·25-s − 0.392·26-s − 2.69·27-s + 1.11·29-s − 0.730·30-s − 0.898·31-s − 2.08·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.152780367\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152780367\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 67 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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
−10.98541347834296746446520843679, −10.44842362966500720226932090160, −9.618538959847183781721636774267, −9.272174820517103042001349727963, −9.032645143961108496014525540660, −8.635654732808729228666065182018, −7.937866989816557168551649806659, −7.66829807172261313648704933573, −7.60937830845796345154928176191, −6.54296788073615386345437574074, −6.45405413344596892238755058393, −5.44825903193512128852002821382, −5.36436203103784840998680545498, −4.68449811645055221549683136102, −4.21766230232826466539162546634, −3.31399820513476425328931648729, −3.23261278261940623949100268241, −2.48155940693625889045567456367, −2.46626714514537242760146275771, −0.44634355792084626236331203684,
0.44634355792084626236331203684, 2.46626714514537242760146275771, 2.48155940693625889045567456367, 3.23261278261940623949100268241, 3.31399820513476425328931648729, 4.21766230232826466539162546634, 4.68449811645055221549683136102, 5.36436203103784840998680545498, 5.44825903193512128852002821382, 6.45405413344596892238755058393, 6.54296788073615386345437574074, 7.60937830845796345154928176191, 7.66829807172261313648704933573, 7.937866989816557168551649806659, 8.635654732808729228666065182018, 9.032645143961108496014525540660, 9.272174820517103042001349727963, 9.618538959847183781721636774267, 10.44842362966500720226932090160, 10.98541347834296746446520843679