| L(s) = 1 | + 2-s − 4-s − 3·5-s + 4·7-s − 3·8-s − 3·10-s + 5·13-s + 4·14-s − 16-s + 3·20-s + 4·25-s + 5·26-s − 4·28-s − 10·29-s + 5·32-s − 12·35-s − 10·37-s + 9·40-s + 10·47-s + 9·49-s + 4·50-s − 5·52-s − 12·56-s − 10·58-s + 7·64-s − 15·65-s + 17·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.34·5-s + 1.51·7-s − 1.06·8-s − 0.948·10-s + 1.38·13-s + 1.06·14-s − 1/4·16-s + 0.670·20-s + 4/5·25-s + 0.980·26-s − 0.755·28-s − 1.85·29-s + 0.883·32-s − 2.02·35-s − 1.64·37-s + 1.42·40-s + 1.45·47-s + 9/7·49-s + 0.565·50-s − 0.693·52-s − 1.60·56-s − 1.31·58-s + 7/8·64-s − 1.86·65-s + 2.07·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.899183875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.899183875\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 18 T + p T^{2} ) \) |
| 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
−8.384907606716604755146909068134, −7.72563785602142736028858164054, −7.45584937274136543385473227213, −7.03495957309274642467837366035, −6.31337650872544824735140754339, −5.75902371009033282955720019688, −5.48507105789302802646487835099, −4.85638567953984695031278672420, −4.56612656321565681987255903759, −3.89945272797370043233009908055, −3.70934164968343504770018970371, −3.30226944818049256621107695580, −2.28364480233662647720881213201, −1.56690003188077209026588135426, −0.62242832475543800281374199005,
0.62242832475543800281374199005, 1.56690003188077209026588135426, 2.28364480233662647720881213201, 3.30226944818049256621107695580, 3.70934164968343504770018970371, 3.89945272797370043233009908055, 4.56612656321565681987255903759, 4.85638567953984695031278672420, 5.48507105789302802646487835099, 5.75902371009033282955720019688, 6.31337650872544824735140754339, 7.03495957309274642467837366035, 7.45584937274136543385473227213, 7.72563785602142736028858164054, 8.384907606716604755146909068134
