L(s) = 1 | + 2·3-s + 4-s − 2·5-s + 9-s − 2·11-s + 2·12-s − 4·15-s + 16-s − 6·17-s − 2·19-s − 2·20-s − 3·25-s + 2·27-s − 4·33-s + 36-s − 12·37-s − 2·43-s − 2·44-s − 2·45-s + 2·48-s − 2·49-s − 12·51-s − 4·53-s + 4·55-s − 4·57-s − 4·60-s + 2·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/2·4-s − 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.577·12-s − 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.447·20-s − 3/5·25-s + 0.384·27-s − 0.696·33-s + 1/6·36-s − 1.97·37-s − 0.304·43-s − 0.301·44-s − 0.298·45-s + 0.288·48-s − 2/7·49-s − 1.68·51-s − 0.549·53-s + 0.539·55-s − 0.529·57-s − 0.516·60-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 229 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.706400781313393545431848402820, −8.374502637742512425496819899751, −7.949304997074426893292933711462, −7.53965544493334983261472688407, −6.87976617567864802160455799887, −6.72216253677178696161679752993, −5.93075277265162786920748460704, −5.31888951878555495821505169743, −4.60763337423274559547691310777, −4.16331239281137423934804170136, −3.47450324154095449914814226183, −3.02271486918722674727643374579, −2.33790921409117387822241258285, −1.75447584848782231422088960947, 0,
1.75447584848782231422088960947, 2.33790921409117387822241258285, 3.02271486918722674727643374579, 3.47450324154095449914814226183, 4.16331239281137423934804170136, 4.60763337423274559547691310777, 5.31888951878555495821505169743, 5.93075277265162786920748460704, 6.72216253677178696161679752993, 6.87976617567864802160455799887, 7.53965544493334983261472688407, 7.949304997074426893292933711462, 8.374502637742512425496819899751, 8.706400781313393545431848402820