| L(s) = 1 | + 38·9-s + 136·11-s − 216·19-s − 332·29-s − 64·31-s + 308·41-s − 49·49-s + 1.72e3·59-s + 780·61-s + 1.68e3·71-s − 2.62e3·79-s + 715·81-s + 1.19e3·89-s + 5.16e3·99-s + 636·101-s + 2.19e3·109-s + 1.12e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.91e3·169-s + ⋯ |
| L(s) = 1 | + 1.40·9-s + 3.72·11-s − 2.60·19-s − 2.12·29-s − 0.370·31-s + 1.17·41-s − 1/7·49-s + 3.79·59-s + 1.63·61-s + 2.80·71-s − 3.73·79-s + 0.980·81-s + 1.42·89-s + 5.24·99-s + 0.626·101-s + 1.92·109-s + 8.42·121-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 + 1.77·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.553942956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.553942956\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3910 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8926 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 p^{2} T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 166 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 p^{2} T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 154 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 114070 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 54498 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 288150 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 860 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 390 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 597926 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 840 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 381134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1312 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 953478 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 598 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 989950 T^{2} + p^{6} T^{4} \) |
| 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
−10.02946915042881982167046298850, −9.914371393876924767852884290849, −9.248316317726779464896581531672, −9.153097173031907510609549520067, −8.549989518073130386600086124067, −8.389544242938031523931893855284, −7.37022809423894979559618856438, −7.17299888115984662670489078459, −6.57370799348107576905041755956, −6.56395588380206983911889159959, −5.97040610034800821680616415276, −5.36217303254421811763986134390, −4.47102922969948699117137854567, −4.08101080907103296914079259256, −3.89755640706809473646846269943, −3.62478676942163227183020315354, −2.13526272177889968834844764803, −1.97004229143794614066581042957, −1.24862246971173620701305384171, −0.65770766187649530306243884125,
0.65770766187649530306243884125, 1.24862246971173620701305384171, 1.97004229143794614066581042957, 2.13526272177889968834844764803, 3.62478676942163227183020315354, 3.89755640706809473646846269943, 4.08101080907103296914079259256, 4.47102922969948699117137854567, 5.36217303254421811763986134390, 5.97040610034800821680616415276, 6.56395588380206983911889159959, 6.57370799348107576905041755956, 7.17299888115984662670489078459, 7.37022809423894979559618856438, 8.389544242938031523931893855284, 8.549989518073130386600086124067, 9.153097173031907510609549520067, 9.248316317726779464896581531672, 9.914371393876924767852884290849, 10.02946915042881982167046298850
