| L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·12-s + 2·13-s + 16-s + 12·17-s + 25-s + 4·27-s − 12·29-s + 3·36-s + 4·39-s − 8·43-s + 2·48-s + 2·49-s + 24·51-s + 2·52-s − 12·53-s − 20·61-s + 64-s + 12·68-s + 2·75-s + 16·79-s + 5·81-s − 24·87-s + 100-s + 36·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 9-s + 0.577·12-s + 0.554·13-s + 1/4·16-s + 2.91·17-s + 1/5·25-s + 0.769·27-s − 2.22·29-s + 1/2·36-s + 0.640·39-s − 1.21·43-s + 0.288·48-s + 2/7·49-s + 3.36·51-s + 0.277·52-s − 1.64·53-s − 2.56·61-s + 1/8·64-s + 1.45·68-s + 0.230·75-s + 1.80·79-s + 5/9·81-s − 2.57·87-s + 1/10·100-s + 3.58·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.116182873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.116182873\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−9.305587122869086271308905422277, −8.925260031303717303381313539625, −7.941231322257043910223245152113, −7.889742230800007366384340287619, −7.70424926324545431494851724983, −6.90106282892871459571015329925, −6.42217617652666865799421983967, −5.66012343286953819079657082523, −5.41509779315100847068653879171, −4.57317770756600442017159530038, −3.70681432856719735440542242113, −3.36585804145949552210873743484, −2.95923023255871163375029903372, −1.87988664142882943456143061246, −1.32801143307487185867201245142,
1.32801143307487185867201245142, 1.87988664142882943456143061246, 2.95923023255871163375029903372, 3.36585804145949552210873743484, 3.70681432856719735440542242113, 4.57317770756600442017159530038, 5.41509779315100847068653879171, 5.66012343286953819079657082523, 6.42217617652666865799421983967, 6.90106282892871459571015329925, 7.70424926324545431494851724983, 7.889742230800007366384340287619, 7.941231322257043910223245152113, 8.925260031303717303381313539625, 9.305587122869086271308905422277
