| L(s) = 1 | − 4·4-s − 7-s − 5·9-s − 11-s + 8·13-s + 12·16-s + 12·17-s − 4·19-s + 6·23-s − 25-s + 4·28-s + 20·36-s + 22·37-s − 12·41-s + 4·44-s + 49-s − 32·52-s − 12·53-s + 20·61-s + 5·63-s − 32·64-s + 10·67-s − 48·68-s + 18·71-s − 4·73-s + 16·76-s + 77-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.377·7-s − 5/3·9-s − 0.301·11-s + 2.21·13-s + 3·16-s + 2.91·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s + 0.755·28-s + 10/3·36-s + 3.61·37-s − 1.87·41-s + 0.603·44-s + 1/7·49-s − 4.43·52-s − 1.64·53-s + 2.56·61-s + 0.629·63-s − 4·64-s + 1.22·67-s − 5.82·68-s + 2.13·71-s − 0.468·73-s + 1.83·76-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456533 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456533 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.099275660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099275660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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
−8.508833833811404945218630536704, −8.196526304421111242297150106130, −8.085327774959512351361821751845, −7.44927930218322722277526119885, −6.46084267575588440942544784885, −5.98016833623442124300848518324, −5.81865396854508245636334597208, −5.22586837194321912962810216772, −4.95929836259470425310107953583, −4.06364144920564711800863579864, −3.61592205748787980286928494613, −3.31516413382692155506861780875, −2.75685822824512438972431635156, −1.19698223664384976616389110079, −0.70813066257818517183339634810,
0.70813066257818517183339634810, 1.19698223664384976616389110079, 2.75685822824512438972431635156, 3.31516413382692155506861780875, 3.61592205748787980286928494613, 4.06364144920564711800863579864, 4.95929836259470425310107953583, 5.22586837194321912962810216772, 5.81865396854508245636334597208, 5.98016833623442124300848518324, 6.46084267575588440942544784885, 7.44927930218322722277526119885, 8.085327774959512351361821751845, 8.196526304421111242297150106130, 8.508833833811404945218630536704
