| 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
−8.467961047217185243736660174457, −7.958882066985845549410362974675, −7.955859097314133791044966496465, −7.36440720073199929079936076774, −6.84651712139757264196140765804, −6.33169466753969442189844266176, −5.51457882926074302977186886250, −5.21730475246735557714050254853, −4.90718026477218821976639724467, −4.13875090005906653574798019221, −3.55640376879290048858690413154, −2.76309813409348720224135008444, −2.51899204061915291020295701141, −1.78111713369696792468537479439, 0,
1.78111713369696792468537479439, 2.51899204061915291020295701141, 2.76309813409348720224135008444, 3.55640376879290048858690413154, 4.13875090005906653574798019221, 4.90718026477218821976639724467, 5.21730475246735557714050254853, 5.51457882926074302977186886250, 6.33169466753969442189844266176, 6.84651712139757264196140765804, 7.36440720073199929079936076774, 7.955859097314133791044966496465, 7.958882066985845549410362974675, 8.467961047217185243736660174457
