| L(s) = 1 | − 8·19-s − 10·25-s + 8·31-s + 2·49-s − 28·61-s + 32·67-s + 20·73-s − 8·79-s + 40·103-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.83·19-s − 2·25-s + 1.43·31-s + 2/7·49-s − 3.58·61-s + 3.90·67-s + 2.34·73-s − 0.900·79-s + 3.94·103-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.353172910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353172910\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.062011207109445203086505410563, −8.555588453497860152279525854744, −8.137073083253253491128724350536, −8.075510122668780086156720606333, −7.51390004933455115710133968594, −7.20558510921975363480825770449, −6.56529551810215557679265974918, −6.30582384584795796831680319876, −6.09561694328624989070662709822, −5.65973933473907663686264215654, −4.90743534700802837146013371229, −4.82723154636183015778244017959, −4.26399833840743041121512714889, −3.74251953912043632090862970789, −3.58764265288227060549944239865, −2.79380017391401252581057271808, −2.21217786533952524625051869433, −2.05525159075940714512799237288, −1.22884542976667090190598622479, −0.38758943591698438846355927388,
0.38758943591698438846355927388, 1.22884542976667090190598622479, 2.05525159075940714512799237288, 2.21217786533952524625051869433, 2.79380017391401252581057271808, 3.58764265288227060549944239865, 3.74251953912043632090862970789, 4.26399833840743041121512714889, 4.82723154636183015778244017959, 4.90743534700802837146013371229, 5.65973933473907663686264215654, 6.09561694328624989070662709822, 6.30582384584795796831680319876, 6.56529551810215557679265974918, 7.20558510921975363480825770449, 7.51390004933455115710133968594, 8.075510122668780086156720606333, 8.137073083253253491128724350536, 8.555588453497860152279525854744, 9.062011207109445203086505410563
