| L(s) = 1 | + 5·3-s + 8·4-s + 16·9-s + 40·12-s + 48·16-s − 49·25-s + 35·27-s + 74·31-s + 128·36-s + 50·37-s + 240·48-s − 98·49-s + 256·64-s − 70·67-s − 245·75-s + 31·81-s + 370·93-s + 190·97-s − 392·100-s − 380·103-s + 280·108-s + 250·111-s + 592·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 5/3·3-s + 2·4-s + 16/9·9-s + 10/3·12-s + 3·16-s − 1.95·25-s + 1.29·27-s + 2.38·31-s + 32/9·36-s + 1.35·37-s + 5·48-s − 2·49-s + 4·64-s − 1.04·67-s − 3.26·75-s + 0.382·81-s + 3.97·93-s + 1.95·97-s − 3.91·100-s − 3.68·103-s + 2.59·108-s + 2.25·111-s + 4.77·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.695766563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.695766563\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 5 T + p^{2} T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )( 1 + 35 T + p^{2} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )( 1 + 50 T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 107 T + p^{2} T^{2} )( 1 + 107 T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 133 T + p^{2} T^{2} )( 1 + 133 T + p^{2} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 95 T + p^{2} T^{2} )^{2} \) |
| 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
−11.40705187730613214962147884641, −11.10753404738016403761117123716, −10.36640137621970247253454740722, −10.02480569420035555321838446259, −9.715510668862999087730019259875, −9.221294436478996552058921796469, −8.326368031668718769837317867305, −8.128504384806102727876891841569, −7.75687021152315944090968060667, −7.35938965037992557991093614051, −6.75144386681675944432626640445, −6.20945163361153676901049389246, −5.98238760160443418989304343754, −4.99063087430731484430696632607, −4.16676985653622901857812369999, −3.60283019630482592464062043948, −2.88002660931729143634468657522, −2.61752170669671940248949119837, −1.91432570704106936968798957549, −1.27822809468157391260321744731,
1.27822809468157391260321744731, 1.91432570704106936968798957549, 2.61752170669671940248949119837, 2.88002660931729143634468657522, 3.60283019630482592464062043948, 4.16676985653622901857812369999, 4.99063087430731484430696632607, 5.98238760160443418989304343754, 6.20945163361153676901049389246, 6.75144386681675944432626640445, 7.35938965037992557991093614051, 7.75687021152315944090968060667, 8.128504384806102727876891841569, 8.326368031668718769837317867305, 9.221294436478996552058921796469, 9.715510668862999087730019259875, 10.02480569420035555321838446259, 10.36640137621970247253454740722, 11.10753404738016403761117123716, 11.40705187730613214962147884641
