L(s) = 1 | − 100·5-s + 3.87e3·9-s − 7.73e5·25-s − 4.16e6·29-s − 3.32e6·41-s − 3.87e5·45-s − 2.15e7·49-s − 4.24e7·61-s − 7.48e7·81-s + 3.22e8·89-s − 3.87e8·101-s + 2.20e8·109-s + 3.87e8·121-s + 1.16e8·125-s + 127-s + 131-s + 137-s + 139-s + 4.16e8·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.52e9·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 0.159·5-s + 0.590·9-s − 1.98·25-s − 5.89·29-s − 1.17·41-s − 0.0945·45-s − 3.74·49-s − 3.06·61-s − 1.73·81-s + 5.14·89-s − 3.71·101-s + 1.56·109-s + 1.80·121-s + 0.477·125-s + 0.942·145-s + 3.09·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.1478786521\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1478786521\) |
\(L(5)\) |
|
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 | $C_2$ | \( ( 1 + 2 p^{2} T + p^{8} T^{2} )^{2} \) |
good | 3 | $C_2^2$ | \( ( 1 - 646 p T^{2} + p^{16} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 220238 p^{2} T^{2} + p^{16} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 193781762 T^{2} + p^{16} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 97037834 p T^{2} + p^{16} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 6043188482 T^{2} + p^{16} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 5539870082 T^{2} + p^{16} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 142069658222 T^{2} + p^{16} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 1041922 T + p^{8} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 487873850882 T^{2} + p^{16} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 3082115005442 T^{2} + p^{16} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 831982 T + p^{8} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 16588024818542 T^{2} + p^{16} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 7011460446062 T^{2} + p^{16} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 7109779173122 T^{2} + p^{16} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 63865039952642 T^{2} + p^{16} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10617778 T + p^{8} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 583910624435218 T^{2} + p^{16} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 207543434244478 T^{2} + p^{16} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 1183233014770562 T^{2} + p^{16} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 336865887770878 T^{2} + p^{16} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 3731242433596142 T^{2} + p^{16} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 80736322 T + p^{8} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 3474443214251522 T^{2} + p^{16} T^{4} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.00083443915422023261798862156, −6.96858691668564559204308352713, −6.34172043198974312739762736923, −6.23757582369929073334200856515, −6.07582074538987587300712214949, −5.75868635530745618459268134434, −5.41071178800638831500692382017, −5.32713873338853142455431870390, −4.92676933425425517985232750969, −4.73269767142320120667910626329, −4.26805590734628580262250456779, −4.12217912254924503675000079622, −3.77472306197706415176602451654, −3.54535807012569873530300333170, −3.42082938970506115434798711658, −2.99836881175627482680437891435, −2.82754981782417286067803685224, −1.99300006853509246466736611099, −1.90905473831419569390846587109, −1.85655848119167621908075159112, −1.59476994122708884207648717350, −1.38583430836009864903264407876, −0.58854728317271444063581659187, −0.40291360473737722197930117533, −0.05872165359954467966744972753,
0.05872165359954467966744972753, 0.40291360473737722197930117533, 0.58854728317271444063581659187, 1.38583430836009864903264407876, 1.59476994122708884207648717350, 1.85655848119167621908075159112, 1.90905473831419569390846587109, 1.99300006853509246466736611099, 2.82754981782417286067803685224, 2.99836881175627482680437891435, 3.42082938970506115434798711658, 3.54535807012569873530300333170, 3.77472306197706415176602451654, 4.12217912254924503675000079622, 4.26805590734628580262250456779, 4.73269767142320120667910626329, 4.92676933425425517985232750969, 5.32713873338853142455431870390, 5.41071178800638831500692382017, 5.75868635530745618459268134434, 6.07582074538987587300712214949, 6.23757582369929073334200856515, 6.34172043198974312739762736923, 6.96858691668564559204308352713, 7.00083443915422023261798862156