L(s) = 1 | − 24·9-s − 56·49-s + 324·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
L(s) = 1 | − 8·9-s − 8·49-s + 36·81-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2535232564\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2535232564\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( ( 1 + p T^{2} )^{4} \) |
good | 3 | \( ( 1 + p T^{2} )^{8} \) |
| 5 | \( ( 1 - 42 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + p T^{2} )^{8} \) |
| 11 | \( ( 1 + 150 T^{4} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - p T^{2} )^{8} \) |
| 17 | \( ( 1 - p T^{2} )^{8} \) |
| 19 | \( ( 1 + 630 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 1770 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 810 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 1110 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 + 7350 T^{4} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 4470 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 12810 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.86435049453839051409941504765, −3.80727114607320931391896797420, −3.54278878327944923712326826428, −3.51543400009347514399929374051, −3.39727092282995342581525988359, −3.38226256236228159172567390728, −3.14800840907423645696040757902, −3.11386387690465826522731801250, −2.98040560144960501088618373655, −2.91559006285694047322890175204, −2.85076866599662300525967976877, −2.68949170469390789328204407063, −2.45119549870087173472430686598, −2.34683113271580161142778332825, −2.29499094838776145664633418840, −2.06025100583167401427033202001, −1.93704521459027060719711839930, −1.69949810424729057754076709054, −1.53735942489445607249577982380, −1.46408388628849698568588690651, −0.909859837434041623540232811268, −0.78545072692815569191357294212, −0.41440683688438305405515625671, −0.28558121446666688944813274362, −0.16841429039001183893974432430,
0.16841429039001183893974432430, 0.28558121446666688944813274362, 0.41440683688438305405515625671, 0.78545072692815569191357294212, 0.909859837434041623540232811268, 1.46408388628849698568588690651, 1.53735942489445607249577982380, 1.69949810424729057754076709054, 1.93704521459027060719711839930, 2.06025100583167401427033202001, 2.29499094838776145664633418840, 2.34683113271580161142778332825, 2.45119549870087173472430686598, 2.68949170469390789328204407063, 2.85076866599662300525967976877, 2.91559006285694047322890175204, 2.98040560144960501088618373655, 3.11386387690465826522731801250, 3.14800840907423645696040757902, 3.38226256236228159172567390728, 3.39727092282995342581525988359, 3.51543400009347514399929374051, 3.54278878327944923712326826428, 3.80727114607320931391896797420, 3.86435049453839051409941504765
Plot not available for L-functions of degree greater than 10.