| L(s) = 1 | + 88·5-s + 2.41e3·13-s + 1.07e4·17-s − 7.88e3·25-s + 2.92e4·37-s + 2.17e5·41-s + 2.66e5·53-s + 7.76e5·61-s + 2.12e5·65-s + 1.07e6·73-s − 5.31e5·81-s + 9.47e5·85-s − 3.93e5·97-s + 3.40e6·101-s − 3.78e6·113-s − 3.54e6·121-s − 2.06e6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.91e6·169-s + ⋯ |
| L(s) = 1 | + 0.703·5-s + 1.09·13-s + 2.19·17-s − 0.504·25-s + 0.577·37-s + 3.15·41-s + 1.79·53-s + 3.42·61-s + 0.773·65-s + 2.77·73-s − 81-s + 1.54·85-s − 0.431·97-s + 3.30·101-s − 2.62·113-s − 2·121-s − 1.05·125-s + 0.603·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(5.062536509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.062536509\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 88 T + p^{6} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{12} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{12} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4070 T + p^{6} T^{2} )( 1 + 1656 T + p^{6} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 9776 T + p^{6} T^{2} )( 1 - 990 T + p^{6} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 31878 T + p^{6} T^{2} )( 1 + 31878 T + p^{6} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 84744 T + p^{6} T^{2} )( 1 + 55510 T + p^{6} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 108560 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 296296 T + p^{6} T^{2} )( 1 + 29430 T + p^{6} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 388440 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{12} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 650016 T + p^{6} T^{2} )( 1 - 427570 T + p^{6} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1378962 T + p^{6} T^{2} )( 1 + 1378962 T + p^{6} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 1078704 T + p^{6} T^{2} )( 1 + 1472510 T + p^{6} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.77826147724802193678895508889, −11.74903926950078118159650607705, −10.90651481536231729961307332732, −10.55828768908469105864198692976, −9.791716052827979083852667487119, −9.725819439488743672562265660110, −9.012873831689916744092270131356, −8.412016659401069092661961509591, −7.81898226038400046764804939245, −7.47050056225287782711781843014, −6.59675133087085575457141375008, −6.05947591637361310994492279519, −5.54478879738374444127527691169, −5.21273802763326736988995812034, −3.91723425199992899462021158853, −3.82596384975116021717640824623, −2.74794547442634824250810588265, −2.15379884661766649270034891376, −1.05311479467472686699189203650, −0.841774997991425983080826553616,
0.841774997991425983080826553616, 1.05311479467472686699189203650, 2.15379884661766649270034891376, 2.74794547442634824250810588265, 3.82596384975116021717640824623, 3.91723425199992899462021158853, 5.21273802763326736988995812034, 5.54478879738374444127527691169, 6.05947591637361310994492279519, 6.59675133087085575457141375008, 7.47050056225287782711781843014, 7.81898226038400046764804939245, 8.412016659401069092661961509591, 9.012873831689916744092270131356, 9.725819439488743672562265660110, 9.791716052827979083852667487119, 10.55828768908469105864198692976, 10.90651481536231729961307332732, 11.74903926950078118159650607705, 11.77826147724802193678895508889
