| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s − 8·11-s + 2·12-s + 16-s + 2·17-s + 3·18-s + 8·19-s − 8·22-s + 2·24-s − 6·25-s + 4·27-s + 32-s − 16·33-s + 2·34-s + 3·36-s + 8·38-s + 20·41-s + 24·43-s − 8·44-s + 2·48-s − 14·49-s − 6·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 9-s − 2.41·11-s + 0.577·12-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 1.83·19-s − 1.70·22-s + 0.408·24-s − 6/5·25-s + 0.769·27-s + 0.176·32-s − 2.78·33-s + 0.342·34-s + 1/2·36-s + 1.29·38-s + 3.12·41-s + 3.65·43-s − 1.20·44-s + 0.288·48-s − 2·49-s − 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.157881714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.157881714\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.658113684153939465660001552715, −7.958882066985845549410362974675, −7.890763986910657579747142309970, −7.36440720073199929079936076774, −7.34292617858622580122915984824, −6.29236664701447308952901966222, −5.62906231402718523510730620943, −5.51457882926074302977186886250, −4.91746541938042376263088347369, −4.13875090005906653574798019221, −3.84610425166553964062381179508, −2.86767294411575254972359726206, −2.76309813409348720224135008444, −2.21434037861733189800471556077, −1.01738437769258744176787321955,
1.01738437769258744176787321955, 2.21434037861733189800471556077, 2.76309813409348720224135008444, 2.86767294411575254972359726206, 3.84610425166553964062381179508, 4.13875090005906653574798019221, 4.91746541938042376263088347369, 5.51457882926074302977186886250, 5.62906231402718523510730620943, 6.29236664701447308952901966222, 7.34292617858622580122915984824, 7.36440720073199929079936076774, 7.890763986910657579747142309970, 7.958882066985845549410362974675, 8.658113684153939465660001552715
