| L(s) = 1 | + 2·3-s + 9-s − 4·19-s − 10·25-s − 4·27-s + 20·43-s + 14·49-s − 8·57-s − 28·67-s + 4·73-s − 20·75-s − 11·81-s − 20·97-s + 14·121-s + 127-s + 40·129-s + 131-s + 137-s + 139-s + 28·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 4·171-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.917·19-s − 2·25-s − 0.769·27-s + 3.04·43-s + 2·49-s − 1.05·57-s − 3.42·67-s + 0.468·73-s − 2.30·75-s − 1.22·81-s − 2.03·97-s + 1.27·121-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.30·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s − 0.305·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.295166367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295166367\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−14.06394328773157206212895907659, −13.83052280484096635574528108503, −13.36478006518469625335286818462, −12.77691477777785881349281164349, −12.10706443513036903495494494042, −11.76916993795214137709269398896, −10.79247496690339236535851258895, −10.63625573446747533218639610596, −9.656089035790711538682577334756, −9.349238772081262739397023984498, −8.727473141562628758708798405880, −8.221852677201233064086912461177, −7.55043553750843323511328883538, −7.18846721052102894550229109732, −5.94979681928227901041780787872, −5.77683474317835016152986578228, −4.35093785569216056385832285131, −3.94892189051276045666934919575, −2.85796536911498613400902789739, −2.05544024252426502892033942734,
2.05544024252426502892033942734, 2.85796536911498613400902789739, 3.94892189051276045666934919575, 4.35093785569216056385832285131, 5.77683474317835016152986578228, 5.94979681928227901041780787872, 7.18846721052102894550229109732, 7.55043553750843323511328883538, 8.221852677201233064086912461177, 8.727473141562628758708798405880, 9.349238772081262739397023984498, 9.656089035790711538682577334756, 10.63625573446747533218639610596, 10.79247496690339236535851258895, 11.76916993795214137709269398896, 12.10706443513036903495494494042, 12.77691477777785881349281164349, 13.36478006518469625335286818462, 13.83052280484096635574528108503, 14.06394328773157206212895907659
