| L(s) = 1 | − 300·9-s − 792·17-s + 1.63e3·25-s + 72·41-s + 6.40e3·49-s − 1.72e4·73-s + 5.43e4·81-s + 1.24e4·89-s + 1.89e4·97-s − 8.56e3·113-s + 1.94e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.37e5·153-s + 157-s + 163-s + 167-s − 1.71e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.70·9-s − 2.74·17-s + 2.61·25-s + 0.0428·41-s + 2.66·49-s − 3.23·73-s + 8.28·81-s + 1.57·89-s + 2.01·97-s − 0.671·113-s + 1.32·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 10.1·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 0.601·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16777216 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16777216 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2539743671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2539743671\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 50 p T^{2} + p^{8} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 818 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 3202 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 9706 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 8590 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 198 T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 196694 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 305666 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1224050 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1029374 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 675890 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 4439894 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3766658 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5132594 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 3526570 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26682482 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 37044950 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 10767742 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4310 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 60727426 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 12229142 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3114 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4730 T + p^{4} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.48208444439372335736088826642, −10.32534132088152752948251025350, −9.481686007732231854472214489392, −9.072037711695866440777488857207, −8.839429104977758911231748579360, −8.811964468252169382442897214326, −8.790310081493905828986506264341, −8.241733772344441412989264341900, −7.933443884584316782843472721459, −7.35012124946592592600709734820, −7.07737409824894396603961043227, −6.65049549890206336722489219775, −6.25581720380942796390978537392, −5.95544391573605105474914829250, −5.81240718264911181139257703898, −5.08360553417680635354609317800, −4.97380559079940507726136771164, −4.55862753770478273440394999282, −3.85439363364656885630509811727, −3.37423326325445336988779573030, −2.69964291033529364057866794096, −2.63060706194808160179673361894, −2.23271083553957618358945910516, −0.959560394294150885479778180248, −0.15894616494435965284829540602,
0.15894616494435965284829540602, 0.959560394294150885479778180248, 2.23271083553957618358945910516, 2.63060706194808160179673361894, 2.69964291033529364057866794096, 3.37423326325445336988779573030, 3.85439363364656885630509811727, 4.55862753770478273440394999282, 4.97380559079940507726136771164, 5.08360553417680635354609317800, 5.81240718264911181139257703898, 5.95544391573605105474914829250, 6.25581720380942796390978537392, 6.65049549890206336722489219775, 7.07737409824894396603961043227, 7.35012124946592592600709734820, 7.933443884584316782843472721459, 8.241733772344441412989264341900, 8.790310081493905828986506264341, 8.811964468252169382442897214326, 8.839429104977758911231748579360, 9.072037711695866440777488857207, 9.481686007732231854472214489392, 10.32534132088152752948251025350, 10.48208444439372335736088826642
