L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 3·9-s + 2·12-s + 16-s − 2·17-s − 3·18-s − 8·19-s − 2·24-s − 10·25-s + 4·27-s − 32-s + 2·34-s + 3·36-s + 8·38-s + 12·41-s − 8·43-s + 2·48-s − 10·49-s + 10·50-s − 4·51-s − 4·54-s − 16·57-s − 24·59-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 + 0.577·12-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 1.83·19-s − 0.408·24-s − 2·25-s + 0.769·27-s − 0.176·32-s + 0.342·34-s + 1/2·36-s + 1.29·38-s + 1.87·41-s − 1.21·43-s + 0.288·48-s − 1.42·49-s + 1.41·50-s − 0.560·51-s − 0.544·54-s − 2.11·57-s − 3.12·59-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 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
−8.367508603382134314146293072792, −8.251849824429749124017602038072, −7.64324124944073722179939518360, −7.50641789037179894281138679970, −6.65576551455226941276409559472, −6.30405633057891562651145051326, −5.96956680213085492365415051586, −5.05586136129239314554307726555, −4.37901858420320549126928147503, −4.05372696693836488551925909574, −3.33704793748910042211032942691, −2.70945693922168405804036190722, −2.04126125287275460099535053657, −1.60962373150081650216951780832, 0,
1.60962373150081650216951780832, 2.04126125287275460099535053657, 2.70945693922168405804036190722, 3.33704793748910042211032942691, 4.05372696693836488551925909574, 4.37901858420320549126928147503, 5.05586136129239314554307726555, 5.96956680213085492365415051586, 6.30405633057891562651145051326, 6.65576551455226941276409559472, 7.50641789037179894281138679970, 7.64324124944073722179939518360, 8.251849824429749124017602038072, 8.367508603382134314146293072792