| L(s) = 1 | − 3-s + 5-s + 4·7-s + 3·9-s + 3·11-s − 12·13-s − 15-s + 5·17-s + 19-s − 4·21-s − 7·23-s + 5·25-s − 8·27-s + 4·29-s − 5·31-s − 3·33-s + 4·35-s − 3·37-s + 12·39-s − 4·41-s + 8·43-s + 3·45-s + 5·47-s + 9·49-s − 5·51-s + 53-s + 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 9-s + 0.904·11-s − 3.32·13-s − 0.258·15-s + 1.21·17-s + 0.229·19-s − 0.872·21-s − 1.45·23-s + 25-s − 1.53·27-s + 0.742·29-s − 0.898·31-s − 0.522·33-s + 0.676·35-s − 0.493·37-s + 1.92·39-s − 0.624·41-s + 1.21·43-s + 0.447·45-s + 0.729·47-s + 9/7·49-s − 0.700·51-s + 0.137·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.068180671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068180671\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−14.26128449298071632015483819022, −13.36637103897624603813564426192, −12.49513974671558037969982310582, −12.36267530916875766472585257593, −11.73197044916876882571546769312, −11.65692809887798795583277751402, −10.57555685519585205276805160244, −10.23119771454001353389512992432, −9.698304938160205601973786242431, −9.376302927091470684983633768693, −8.379656881043203736225585759727, −7.74974602213413377460865354291, −7.14043922179287005866057068800, −7.00594909735142233711115444878, −5.57360598038331244081150557594, −5.44255037315216003872634593352, −4.59725453899486130670035939670, −4.11761957989625921924448635341, −2.52824303702105238294123305358, −1.60478291472805514387162796273,
1.60478291472805514387162796273, 2.52824303702105238294123305358, 4.11761957989625921924448635341, 4.59725453899486130670035939670, 5.44255037315216003872634593352, 5.57360598038331244081150557594, 7.00594909735142233711115444878, 7.14043922179287005866057068800, 7.74974602213413377460865354291, 8.379656881043203736225585759727, 9.376302927091470684983633768693, 9.698304938160205601973786242431, 10.23119771454001353389512992432, 10.57555685519585205276805160244, 11.65692809887798795583277751402, 11.73197044916876882571546769312, 12.36267530916875766472585257593, 12.49513974671558037969982310582, 13.36637103897624603813564426192, 14.26128449298071632015483819022
