| L(s) = 1 | − 9·9-s − 108·11-s + 50·19-s + 108·29-s − 542·31-s − 720·41-s + 637·49-s − 252·59-s + 94·61-s − 2.16e3·71-s + 1.13e3·79-s + 81·81-s − 2.88e3·89-s + 972·99-s + 1.65e3·101-s − 2.55e3·109-s + 6.08e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.36e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.96·11-s + 0.603·19-s + 0.691·29-s − 3.14·31-s − 2.74·41-s + 13/7·49-s − 0.556·59-s + 0.197·61-s − 3.61·71-s + 1.61·79-s + 1/9·81-s − 3.43·89-s + 0.986·99-s + 1.63·101-s − 2.24·109-s + 4.57·121-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 + 0.623·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2483238735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2483238735\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 13 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1369 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9502 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 25 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 271 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2710 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 132445 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 64838 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 296458 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 483877 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1080 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 332882 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 568 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 878510 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1440 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1632625 T^{2} + p^{6} T^{4} \) |
| 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
−11.60609884591652711900355878266, −10.75672836086343634244033158448, −10.70637666713420104915664190723, −10.29633388406854465694486111123, −9.783198211330513501265108725625, −9.126702531666962289188497627986, −8.634771083437994388019404849359, −8.189119290841646359411591304701, −7.67795466423533824726446588063, −7.25619094007850076495064708602, −6.87690304715967244055799843585, −5.69424975341897271626081421592, −5.62510119856003649197402528406, −5.14037377627294060158795383127, −4.53140699414763792052279030522, −3.55770088522406099045002116059, −3.00958633973042921810540071001, −2.42987144276404713890696522463, −1.62210006092670562822078350465, −0.17602046240831288272219351373,
0.17602046240831288272219351373, 1.62210006092670562822078350465, 2.42987144276404713890696522463, 3.00958633973042921810540071001, 3.55770088522406099045002116059, 4.53140699414763792052279030522, 5.14037377627294060158795383127, 5.62510119856003649197402528406, 5.69424975341897271626081421592, 6.87690304715967244055799843585, 7.25619094007850076495064708602, 7.67795466423533824726446588063, 8.189119290841646359411591304701, 8.634771083437994388019404849359, 9.126702531666962289188497627986, 9.783198211330513501265108725625, 10.29633388406854465694486111123, 10.70637666713420104915664190723, 10.75672836086343634244033158448, 11.60609884591652711900355878266
