L(s) = 1 | + 3·2-s + 4·3-s + 4·4-s + 3·5-s + 12·6-s + 3·8-s + 6·9-s + 9·10-s + 16·12-s − 5·13-s + 12·15-s + 3·16-s + 3·17-s + 18·18-s + 12·20-s + 6·23-s + 12·24-s + 25-s − 15·26-s − 4·27-s − 3·29-s + 36·30-s + 6·31-s + 6·32-s + 9·34-s + 24·36-s − 15·37-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 2.30·3-s + 2·4-s + 1.34·5-s + 4.89·6-s + 1.06·8-s + 2·9-s + 2.84·10-s + 4.61·12-s − 1.38·13-s + 3.09·15-s + 3/4·16-s + 0.727·17-s + 4.24·18-s + 2.68·20-s + 1.25·23-s + 2.44·24-s + 1/5·25-s − 2.94·26-s − 0.769·27-s − 0.557·29-s + 6.57·30-s + 1.07·31-s + 1.06·32-s + 1.54·34-s + 4·36-s − 2.46·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(14.00015618\) |
\(L(\frac12)\) |
\(\approx\) |
\(14.00015618\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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
−10.56458074046539108975313031553, −10.35628436306160952597990326764, −9.716522843062953201622080943076, −9.618805000365978422066030827903, −8.868811495633391738716357431185, −8.866549478788611556476117902161, −7.951297574699014768279554613630, −7.945662518703252913647225399735, −7.00192994302213004382705140619, −6.94950855701763413424743705058, −6.01467022436698410057546774281, −5.62078077942554302737920126240, −5.11672081659661309344851513405, −4.90163249984040425247488313072, −4.13525514619251987318892384868, −3.50340197619192825366881862776, −3.13130836712315482877471682842, −2.89974443072868852318607289491, −2.03865981298108413502466746698, −1.82131636698860705670542706764,
1.82131636698860705670542706764, 2.03865981298108413502466746698, 2.89974443072868852318607289491, 3.13130836712315482877471682842, 3.50340197619192825366881862776, 4.13525514619251987318892384868, 4.90163249984040425247488313072, 5.11672081659661309344851513405, 5.62078077942554302737920126240, 6.01467022436698410057546774281, 6.94950855701763413424743705058, 7.00192994302213004382705140619, 7.945662518703252913647225399735, 7.951297574699014768279554613630, 8.866549478788611556476117902161, 8.868811495633391738716357431185, 9.618805000365978422066030827903, 9.716522843062953201622080943076, 10.35628436306160952597990326764, 10.56458074046539108975313031553