| L(s) = 1 | − 3·9-s − 6·11-s + 72·23-s + 47·25-s − 102·29-s + 44·37-s + 20·43-s + 102·53-s + 136·67-s + 250·79-s + 9·81-s + 18·99-s − 66·107-s − 64·109-s − 372·113-s − 215·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 290·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 0.545·11-s + 3.13·23-s + 1.87·25-s − 3.51·29-s + 1.18·37-s + 0.465·43-s + 1.92·53-s + 2.02·67-s + 3.16·79-s + 1/9·81-s + 2/11·99-s − 0.616·107-s − 0.587·109-s − 3.29·113-s − 1.77·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.71·169-s + 0.00578·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.272212213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272212213\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 47 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 278 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 614 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 36 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 51 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1775 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2774 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3694 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 51 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 1415 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2642 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 10226 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 125 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9985 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 10550 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2857 T^{2} + p^{4} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.88484581889301396937773275002, −10.43457411803844137389744394665, −9.680494070722267451736399999412, −9.330449821790165089067214546887, −8.900985989644728305812018539998, −8.805827613920809630482659868080, −7.86420281077274157589746719140, −7.70964950945993624050986492244, −7.02359611513329886255111438633, −6.85019895625701198123414534850, −6.20223034713417550531271925703, −5.44368997075712284515016128735, −5.16439083935614216247502912046, −4.93022628407194996076895031433, −3.84082722727212344194645525780, −3.64638313430677821621083119574, −2.64447887327741477051064543753, −2.53134148614551778941647434480, −1.32700219554141377948188496052, −0.61990055605591198295575321979,
0.61990055605591198295575321979, 1.32700219554141377948188496052, 2.53134148614551778941647434480, 2.64447887327741477051064543753, 3.64638313430677821621083119574, 3.84082722727212344194645525780, 4.93022628407194996076895031433, 5.16439083935614216247502912046, 5.44368997075712284515016128735, 6.20223034713417550531271925703, 6.85019895625701198123414534850, 7.02359611513329886255111438633, 7.70964950945993624050986492244, 7.86420281077274157589746719140, 8.805827613920809630482659868080, 8.900985989644728305812018539998, 9.330449821790165089067214546887, 9.680494070722267451736399999412, 10.43457411803844137389744394665, 10.88484581889301396937773275002
