| L(s) = 1 | − 8·5-s + 2·7-s − 11·9-s − 28·11-s + 46·17-s − 20·19-s + 2·23-s − 2·25-s − 16·35-s − 136·43-s + 88·45-s − 52·47-s − 95·49-s + 224·55-s − 80·61-s − 22·63-s − 14·73-s − 56·77-s + 40·81-s − 64·83-s − 368·85-s + 160·95-s + 308·99-s + 28·101-s − 16·115-s + 92·119-s + 346·121-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 2/7·7-s − 1.22·9-s − 2.54·11-s + 2.70·17-s − 1.05·19-s + 2/23·23-s − 0.0799·25-s − 0.457·35-s − 3.16·43-s + 1.95·45-s − 1.10·47-s − 1.93·49-s + 4.07·55-s − 1.31·61-s − 0.349·63-s − 0.191·73-s − 0.727·77-s + 0.493·81-s − 0.771·83-s − 4.32·85-s + 1.68·95-s + 28/9·99-s + 0.277·101-s − 0.139·115-s + 0.773·119-s + 2.85·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.08347711349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08347711349\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 20 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 11 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 77 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 p T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 878 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1694 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2318 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 68 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 907 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6701 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8717 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9038 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3086 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 862 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 9422 T^{2} + p^{4} T^{4} \) |
| 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
−11.92291021531373761057622766743, −11.51089124983979288171443922154, −10.74081752123605529282444153474, −10.45755220129903749774002090404, −9.976499739868240122773070602933, −9.529168209752329272359064676833, −8.429720045638069011878038389143, −8.268795072332655496933284078998, −7.933928393804037235717706352640, −7.76197182528621233414732282366, −7.10142279471633627440903028542, −6.21014687627168187753861127816, −5.67758016848150475117776113762, −5.07188702153644665249628450593, −4.87069850791155669445467921996, −3.82771571257552130562890958955, −3.08151311742406004573393876917, −3.05436377462331923120413964721, −1.77142759476623953963895995082, −0.13452042665839393995238579523,
0.13452042665839393995238579523, 1.77142759476623953963895995082, 3.05436377462331923120413964721, 3.08151311742406004573393876917, 3.82771571257552130562890958955, 4.87069850791155669445467921996, 5.07188702153644665249628450593, 5.67758016848150475117776113762, 6.21014687627168187753861127816, 7.10142279471633627440903028542, 7.76197182528621233414732282366, 7.933928393804037235717706352640, 8.268795072332655496933284078998, 8.429720045638069011878038389143, 9.529168209752329272359064676833, 9.976499739868240122773070602933, 10.45755220129903749774002090404, 10.74081752123605529282444153474, 11.51089124983979288171443922154, 11.92291021531373761057622766743
