| L(s) = 1 | + 2·2-s + 3-s − 4-s + 5-s + 2·6-s − 8·8-s + 9-s + 2·10-s − 12-s + 4·13-s + 15-s − 7·16-s + 2·18-s − 20-s − 8·24-s + 25-s + 8·26-s + 27-s + 2·30-s + 14·32-s − 36-s + 4·39-s − 8·40-s + 20·41-s + 45-s − 7·48-s − 7·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 2.82·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s + 1.10·13-s + 0.258·15-s − 7/4·16-s + 0.471·18-s − 0.223·20-s − 1.63·24-s + 1/5·25-s + 1.56·26-s + 0.192·27-s + 0.365·30-s + 2.47·32-s − 1/6·36-s + 0.640·39-s − 1.26·40-s + 3.12·41-s + 0.149·45-s − 1.01·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.063059717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.063059717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.890589428908360265391160029852, −8.843956595114680827452383410006, −8.627000477337622059216224790867, −7.68003844385765985083943135134, −7.37485778338999061621083536758, −6.39356342307905775022074491331, −5.91397914498304472267826736485, −5.86497653842965752491328535183, −5.03119182641673168471146566272, −4.52160444884874669934496061646, −4.09922216243454775532088496022, −3.56571005686312005410318267112, −3.01549784140686353642765162463, −2.32940333880865099555943992805, −0.964820104627525128067217267144,
0.964820104627525128067217267144, 2.32940333880865099555943992805, 3.01549784140686353642765162463, 3.56571005686312005410318267112, 4.09922216243454775532088496022, 4.52160444884874669934496061646, 5.03119182641673168471146566272, 5.86497653842965752491328535183, 5.91397914498304472267826736485, 6.39356342307905775022074491331, 7.37485778338999061621083536758, 7.68003844385765985083943135134, 8.627000477337622059216224790867, 8.843956595114680827452383410006, 8.890589428908360265391160029852
