| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 8·11-s − 12-s − 4·13-s + 16-s − 18-s − 8·22-s + 24-s − 6·25-s + 4·26-s − 27-s − 32-s − 8·33-s + 36-s − 4·37-s + 4·39-s + 8·44-s − 48-s − 14·49-s + 6·50-s − 4·52-s + 54-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 2.41·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.235·18-s − 1.70·22-s + 0.204·24-s − 6/5·25-s + 0.784·26-s − 0.192·27-s − 0.176·32-s − 1.39·33-s + 1/6·36-s − 0.657·37-s + 0.640·39-s + 1.20·44-s − 0.144·48-s − 2·49-s + 0.848·50-s − 0.554·52-s + 0.136·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−9.080746654144001493159861980940, −8.000937863938982294341146593643, −7.972624440390951709752181868622, −7.22142790948264379459326388401, −6.79533660929922109224513153399, −6.37869009161194820234692371426, −6.05886413503827459123865903257, −5.36244298721167160718725340278, −4.66119827871442481374467622682, −4.24541437163957543556604196065, −3.56656081546400251035965478912, −2.92323665285260202872977027719, −1.78155952274398559703453953653, −1.43438626728056627432143840048, 0,
1.43438626728056627432143840048, 1.78155952274398559703453953653, 2.92323665285260202872977027719, 3.56656081546400251035965478912, 4.24541437163957543556604196065, 4.66119827871442481374467622682, 5.36244298721167160718725340278, 6.05886413503827459123865903257, 6.37869009161194820234692371426, 6.79533660929922109224513153399, 7.22142790948264379459326388401, 7.972624440390951709752181868622, 8.000937863938982294341146593643, 9.080746654144001493159861980940
