L(s) = 1 | − 2-s − 4-s − 4·7-s + 3·8-s − 2·9-s + 4·11-s + 4·14-s − 16-s + 2·18-s − 4·22-s + 2·25-s + 4·28-s − 4·31-s − 5·32-s + 2·36-s − 4·41-s − 4·44-s + 2·49-s − 2·50-s − 12·56-s + 4·61-s + 4·62-s + 8·63-s + 7·64-s − 6·72-s − 16·77-s − 5·81-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.51·7-s + 1.06·8-s − 2/3·9-s + 1.20·11-s + 1.06·14-s − 1/4·16-s + 0.471·18-s − 0.852·22-s + 2/5·25-s + 0.755·28-s − 0.718·31-s − 0.883·32-s + 1/3·36-s − 0.624·41-s − 0.603·44-s + 2/7·49-s − 0.282·50-s − 1.60·56-s + 0.512·61-s + 0.508·62-s + 1.00·63-s + 7/8·64-s − 0.707·72-s − 1.82·77-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102152 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102152 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 113 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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
−9.348396108882065590660561067358, −8.795915229466260837721403932826, −8.642266317389829838755969135469, −7.932123018820327893242468220937, −7.30311577361270594907674870701, −6.83601205444564967473881300740, −6.34459537101141384316686845651, −5.85482207246280021627662883848, −5.14765529784341833324469307076, −4.51031323860588565125821056432, −3.64770643662961912713630044636, −3.46115980038854906462600056276, −2.44055725229160967607899501470, −1.27940014528468838562905924997, 0,
1.27940014528468838562905924997, 2.44055725229160967607899501470, 3.46115980038854906462600056276, 3.64770643662961912713630044636, 4.51031323860588565125821056432, 5.14765529784341833324469307076, 5.85482207246280021627662883848, 6.34459537101141384316686845651, 6.83601205444564967473881300740, 7.30311577361270594907674870701, 7.932123018820327893242468220937, 8.642266317389829838755969135469, 8.795915229466260837721403932826, 9.348396108882065590660561067358