L(s) = 1 | − 16·4-s − 34·5-s − 480·13-s + 256·16-s − 322·17-s + 544·20-s + 578·25-s − 1.59e3·29-s − 3.84e3·37-s − 1.59e3·41-s + 7.68e3·52-s + 4.31e3·61-s − 4.09e3·64-s + 1.63e4·65-s + 5.15e3·68-s + 1.20e4·73-s − 8.70e3·80-s − 6.56e3·81-s + 1.09e4·85-s − 2.49e4·89-s + 2.06e4·97-s − 9.24e3·100-s + 1.58e4·101-s + 1.24e4·109-s + 1.79e4·113-s + 2.55e4·116-s − 2.12e4·125-s + ⋯ |
L(s) = 1 | − 4-s − 1.35·5-s − 2.84·13-s + 16-s − 1.11·17-s + 1.35·20-s + 0.924·25-s − 1.90·29-s − 2.80·37-s − 0.950·41-s + 2.84·52-s + 1.16·61-s − 64-s + 3.86·65-s + 1.11·68-s + 2.25·73-s − 1.35·80-s − 81-s + 1.51·85-s − 3.15·89-s + 2.19·97-s − 0.924·100-s + 1.55·101-s + 1.05·109-s + 1.40·113-s + 1.90·116-s − 1.35·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0005194433078\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0005194433078\) |
\(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 17 | $C_2$ | \( 1 + 322 T + p^{4} T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 14 T + p^{4} T^{2} )( 1 + 48 T + p^{4} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 240 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 82 T + p^{4} T^{2} )( 1 + 1680 T + p^{4} T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 1680 T + p^{4} T^{2} )( 1 + 2162 T + p^{4} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 1440 T + p^{4} T^{2} )( 1 + 3038 T + p^{4} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2482 T + p^{4} T^{2} )( 1 + 2482 T + p^{4} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6958 T + p^{4} T^{2} )( 1 + 2640 T + p^{4} T^{2} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10560 T + p^{4} T^{2} )( 1 - 1442 T + p^{4} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12480 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18720 T + p^{4} T^{2} )( 1 - 1918 T + p^{4} T^{2} ) \) |
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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
−14.97152954424649128334358495527, −14.10419492835511078713885733152, −12.94726882385079371887987361012, −12.68239077092362601332805594278, −12.17559914991501170005584992506, −11.61009834454955228349232683276, −11.08607081626559005411674773584, −9.996998860814354939321159463853, −9.987440899112732180904059164062, −8.908281549867121101570835669852, −8.680889549002433773658074919595, −7.64740205512944680683115550140, −7.41577496693354809507373937242, −6.72843344538868948787304980904, −5.21830414548670572102105733607, −5.01603649752247585741162261375, −4.12681585531230022758155061165, −3.43980326353909428805831619891, −2.15686454539369302130736498381, −0.01186975090612976958539187239,
0.01186975090612976958539187239, 2.15686454539369302130736498381, 3.43980326353909428805831619891, 4.12681585531230022758155061165, 5.01603649752247585741162261375, 5.21830414548670572102105733607, 6.72843344538868948787304980904, 7.41577496693354809507373937242, 7.64740205512944680683115550140, 8.680889549002433773658074919595, 8.908281549867121101570835669852, 9.987440899112732180904059164062, 9.996998860814354939321159463853, 11.08607081626559005411674773584, 11.61009834454955228349232683276, 12.17559914991501170005584992506, 12.68239077092362601332805594278, 12.94726882385079371887987361012, 14.10419492835511078713885733152, 14.97152954424649128334358495527