| L(s) = 1 | − 4·4-s + 6·5-s + 7-s + 12·16-s + 6·17-s − 24·20-s + 17·25-s − 4·28-s + 6·35-s − 14·37-s − 6·41-s − 2·43-s + 18·47-s + 49-s + 18·59-s − 32·64-s − 8·67-s − 24·68-s − 2·79-s + 72·80-s + 6·83-s + 36·85-s + 12·89-s − 68·100-s + 12·101-s + 22·109-s + 12·112-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.68·5-s + 0.377·7-s + 3·16-s + 1.45·17-s − 5.36·20-s + 17/5·25-s − 0.755·28-s + 1.01·35-s − 2.30·37-s − 0.937·41-s − 0.304·43-s + 2.62·47-s + 1/7·49-s + 2.34·59-s − 4·64-s − 0.977·67-s − 2.91·68-s − 0.225·79-s + 8.04·80-s + 0.658·83-s + 3.90·85-s + 1.27·89-s − 6.79·100-s + 1.19·101-s + 2.10·109-s + 1.13·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250047 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250047 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030580821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030580821\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.905165354580512836683090859524, −8.677781718759589525261607373664, −8.386609491702580500148378137845, −7.46477788862935127887709071190, −7.19528682144870768370486195193, −6.04902466517769458594559190180, −6.04248676207121526761624269951, −5.40863841069341263955171468202, −5.13832633779419862585354738180, −4.79831898195110296666259349467, −3.79566092998735399945677628017, −3.44521260796395376114982703678, −2.42168483312985269777609133770, −1.69414023663856774565794959548, −0.980257339797202376885739263498,
0.980257339797202376885739263498, 1.69414023663856774565794959548, 2.42168483312985269777609133770, 3.44521260796395376114982703678, 3.79566092998735399945677628017, 4.79831898195110296666259349467, 5.13832633779419862585354738180, 5.40863841069341263955171468202, 6.04248676207121526761624269951, 6.04902466517769458594559190180, 7.19528682144870768370486195193, 7.46477788862935127887709071190, 8.386609491702580500148378137845, 8.677781718759589525261607373664, 8.905165354580512836683090859524
