L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 6·13-s + 16-s + 2·17-s + 6·23-s + 9·25-s − 4·27-s + 18·29-s − 3·36-s − 12·39-s + 14·43-s − 2·48-s − 49-s − 4·51-s − 6·52-s − 20·53-s + 22·61-s − 64-s − 2·68-s − 12·69-s − 18·75-s − 24·79-s + 5·81-s − 36·87-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1.66·13-s + 1/4·16-s + 0.485·17-s + 1.25·23-s + 9/5·25-s − 0.769·27-s + 3.34·29-s − 1/2·36-s − 1.92·39-s + 2.13·43-s − 0.288·48-s − 1/7·49-s − 0.560·51-s − 0.832·52-s − 2.74·53-s + 2.81·61-s − 1/8·64-s − 0.242·68-s − 1.44·69-s − 2.07·75-s − 2.70·79-s + 5/9·81-s − 3.85·87-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
\(L(1)\) |
\(\approx\) |
\(1.409753685\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409753685\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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
−11.04505362659265251242170856084, −10.78104301170401505307511686094, −10.03956400679909049827172074353, −10.02004710064398676615273167938, −9.219480901588857273058422341604, −8.783531309560911449799472382690, −8.383785026984063472299012064678, −8.109639533376156155836635696537, −7.13626991888380856938724060512, −6.98106462795789150955807148679, −6.23283583615769451213451930779, −6.16423625528239770109262574331, −5.42033514195581625818776081159, −4.89211644729518278801671489593, −4.58595412521107281643708288977, −3.98271519982292327097434245597, −3.18728483142858966879703378545, −2.72104849752177574724365641659, −1.11881376874371552172747247000, −1.03407213013071041944117832355,
1.03407213013071041944117832355, 1.11881376874371552172747247000, 2.72104849752177574724365641659, 3.18728483142858966879703378545, 3.98271519982292327097434245597, 4.58595412521107281643708288977, 4.89211644729518278801671489593, 5.42033514195581625818776081159, 6.16423625528239770109262574331, 6.23283583615769451213451930779, 6.98106462795789150955807148679, 7.13626991888380856938724060512, 8.109639533376156155836635696537, 8.383785026984063472299012064678, 8.783531309560911449799472382690, 9.219480901588857273058422341604, 10.02004710064398676615273167938, 10.03956400679909049827172074353, 10.78104301170401505307511686094, 11.04505362659265251242170856084