| L(s) = 1 | − 9·9-s + 44·11-s + 106·19-s − 44·29-s − 70·31-s − 936·41-s + 325·49-s − 892·59-s + 254·61-s + 72·71-s − 2.73e3·79-s + 81·81-s − 288·89-s − 396·99-s − 2.88e3·101-s − 1.40e3·109-s − 1.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.20·11-s + 1.27·19-s − 0.281·29-s − 0.405·31-s − 3.56·41-s + 0.947·49-s − 1.96·59-s + 0.533·61-s + 0.120·71-s − 3.89·79-s + 1/9·81-s − 0.343·89-s − 0.402·99-s − 2.83·101-s − 1.23·109-s − 0.909·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.99·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.862483826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862483826\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 325 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4393 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6462 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 53 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 35 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 28406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 468 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 26747 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 154746 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 446 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 127 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 56195 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 505550 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1368 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 151470 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 144 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 661105 T^{2} + p^{6} T^{4} \) |
| 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
−10.53158371058046218806584971044, −9.877634234738837143114867484301, −9.722843475057498007671338704347, −9.031308307153737478236886534990, −8.914438510187738876439866311626, −8.263014615725009013937440777588, −7.947404079569868185136553003981, −7.17506162397990181815603387619, −6.99378367909157846798270436601, −6.50032104488283399902715301273, −5.92953983437766620278930323305, −5.27622247064438204674753108195, −5.19424292580739364108337406882, −4.15529490822911756044975170720, −3.99724911230310481174764778629, −3.04357003014518865393587486020, −2.97583692251408573816707361094, −1.59624120750350858608660224521, −1.55241600458236557118752941091, −0.40343040447119746235584574703,
0.40343040447119746235584574703, 1.55241600458236557118752941091, 1.59624120750350858608660224521, 2.97583692251408573816707361094, 3.04357003014518865393587486020, 3.99724911230310481174764778629, 4.15529490822911756044975170720, 5.19424292580739364108337406882, 5.27622247064438204674753108195, 5.92953983437766620278930323305, 6.50032104488283399902715301273, 6.99378367909157846798270436601, 7.17506162397990181815603387619, 7.947404079569868185136553003981, 8.263014615725009013937440777588, 8.914438510187738876439866311626, 9.031308307153737478236886534990, 9.722843475057498007671338704347, 9.877634234738837143114867484301, 10.53158371058046218806584971044
