| L(s) = 1 | + 3-s + 4-s + 4·7-s + 9-s + 12-s − 8·13-s + 16-s − 8·19-s + 4·21-s − 10·25-s + 27-s + 4·28-s + 16·31-s + 36-s − 20·37-s − 8·39-s + 16·43-s + 48-s − 2·49-s − 8·52-s − 8·57-s + 16·61-s + 4·63-s + 64-s − 8·67-s + 4·73-s − 10·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s − 1.83·19-s + 0.872·21-s − 2·25-s + 0.192·27-s + 0.755·28-s + 2.87·31-s + 1/6·36-s − 3.28·37-s − 1.28·39-s + 2.43·43-s + 0.144·48-s − 2/7·49-s − 1.10·52-s − 1.05·57-s + 2.04·61-s + 0.503·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s − 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13068 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13068 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.439066996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.439066996\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−11.31231841896418545254169443203, −10.60919551346140607612352707183, −10.21971960261641202232640310225, −9.723620681626334246237602260556, −8.958855021658912251315559727512, −8.215106046446206217336510515309, −8.013104010237496671597514230187, −7.37562026200620347083900898682, −6.77506565559004391547494912024, −6.01365350861836338965349792477, −4.96524457370890568674874182561, −4.69333471332706230755932945396, −3.77672609499705967583268412266, −2.37545091357255938865951638025, −2.06167323195037554066088312898,
2.06167323195037554066088312898, 2.37545091357255938865951638025, 3.77672609499705967583268412266, 4.69333471332706230755932945396, 4.96524457370890568674874182561, 6.01365350861836338965349792477, 6.77506565559004391547494912024, 7.37562026200620347083900898682, 8.013104010237496671597514230187, 8.215106046446206217336510515309, 8.958855021658912251315559727512, 9.723620681626334246237602260556, 10.21971960261641202232640310225, 10.60919551346140607612352707183, 11.31231841896418545254169443203
