| L(s) = 1 | + 2·3-s − 3·4-s − 4·5-s + 3·9-s + 4·11-s − 6·12-s − 8·15-s + 5·16-s + 12·20-s + 2·25-s + 4·27-s + 8·33-s − 9·36-s + 12·37-s − 12·44-s − 12·45-s + 10·48-s + 49-s + 12·53-s − 16·55-s + 24·59-s + 24·60-s − 3·64-s + 8·67-s + 4·75-s − 20·80-s + 5·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s − 1.78·5-s + 9-s + 1.20·11-s − 1.73·12-s − 2.06·15-s + 5/4·16-s + 2.68·20-s + 2/5·25-s + 0.769·27-s + 1.39·33-s − 3/2·36-s + 1.97·37-s − 1.80·44-s − 1.78·45-s + 1.44·48-s + 1/7·49-s + 1.64·53-s − 2.15·55-s + 3.12·59-s + 3.09·60-s − 3/8·64-s + 0.977·67-s + 0.461·75-s − 2.23·80-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.039703896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039703896\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−9.724661210726173973340366981146, −9.563733444203339339566954721491, −8.699165629902587420835899742101, −8.672365196683144779961098572501, −8.166185648107227108757068218850, −7.59860530511322577039011534740, −7.25047783802838427330028450541, −6.50104795223760620212837147128, −5.62767376699147437191457045340, −4.79135217072233780435269822604, −4.13559084050773741974089362056, −3.91805277407899456693617223448, −3.56599684396412385637222444746, −2.43470322830479280229086031798, −0.860569456173605054070248949719,
0.860569456173605054070248949719, 2.43470322830479280229086031798, 3.56599684396412385637222444746, 3.91805277407899456693617223448, 4.13559084050773741974089362056, 4.79135217072233780435269822604, 5.62767376699147437191457045340, 6.50104795223760620212837147128, 7.25047783802838427330028450541, 7.59860530511322577039011534740, 8.166185648107227108757068218850, 8.672365196683144779961098572501, 8.699165629902587420835899742101, 9.563733444203339339566954721491, 9.724661210726173973340366981146
