| L(s) = 1 | + 4·5-s − 2·7-s − 9-s + 6·11-s + 8·13-s − 10·17-s − 10·19-s − 2·23-s + 11·25-s + 10·29-s − 8·35-s + 16·37-s − 4·43-s − 4·45-s + 10·47-s + 2·49-s + 24·55-s − 10·59-s + 2·61-s + 2·63-s + 32·65-s + 4·67-s − 10·73-s − 12·77-s − 16·79-s + 81-s − 40·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s − 1/3·9-s + 1.80·11-s + 2.21·13-s − 2.42·17-s − 2.29·19-s − 0.417·23-s + 11/5·25-s + 1.85·29-s − 1.35·35-s + 2.63·37-s − 0.609·43-s − 0.596·45-s + 1.45·47-s + 2/7·49-s + 3.23·55-s − 1.30·59-s + 0.256·61-s + 0.251·63-s + 3.96·65-s + 0.488·67-s − 1.17·73-s − 1.36·77-s − 1.80·79-s + 1/9·81-s − 4.33·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.639878196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.639878196\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 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
−9.256473881947371612073563219878, −8.990444829214126718087643036962, −8.812205023461621780437310772633, −8.401492753031991004399717839328, −8.180617586593122847215805049213, −7.16336271125217973537205697222, −6.62625680537001582719136638848, −6.55542940949680066869003220827, −6.15795237426780743743191334314, −6.12073551071207084446192241242, −5.77010002388833548594671666715, −4.79530082223550753269858816088, −4.29505691456533984727706142165, −4.22875992285497761893299684387, −3.67555580224816900608668830724, −2.75524336762364591541307244698, −2.60445532206673706419075594005, −1.87864974831511643511132919400, −1.46850488221194503375374315163, −0.71314714966815865110588352838,
0.71314714966815865110588352838, 1.46850488221194503375374315163, 1.87864974831511643511132919400, 2.60445532206673706419075594005, 2.75524336762364591541307244698, 3.67555580224816900608668830724, 4.22875992285497761893299684387, 4.29505691456533984727706142165, 4.79530082223550753269858816088, 5.77010002388833548594671666715, 6.12073551071207084446192241242, 6.15795237426780743743191334314, 6.55542940949680066869003220827, 6.62625680537001582719136638848, 7.16336271125217973537205697222, 8.180617586593122847215805049213, 8.401492753031991004399717839328, 8.812205023461621780437310772633, 8.990444829214126718087643036962, 9.256473881947371612073563219878
