| L(s) = 1 | + 2·5-s − 9-s − 6·11-s − 25-s − 16·29-s − 8·31-s − 6·41-s − 2·45-s + 5·49-s − 12·55-s + 8·59-s + 18·61-s − 14·71-s − 10·79-s + 81-s − 22·89-s + 6·99-s − 4·101-s + 32·109-s + 5·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 32·145-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s − 1.80·11-s − 1/5·25-s − 2.97·29-s − 1.43·31-s − 0.937·41-s − 0.298·45-s + 5/7·49-s − 1.61·55-s + 1.04·59-s + 2.30·61-s − 1.66·71-s − 1.12·79-s + 1/9·81-s − 2.33·89-s + 0.603·99-s − 0.398·101-s + 3.06·109-s + 5/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.65·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3694496080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3694496080\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−8.895136151705939469094267602375, −8.690765611090871338018468546614, −7.925169463819268935943827772335, −7.84185637652786115042585901373, −7.45180014643452088743916269681, −6.92142553564184139105886482795, −6.85043000871931815896342313727, −5.92227696453935993708262691378, −5.81127752381652632262207622876, −5.37159320041324736889636203218, −5.36656148623248625541468556729, −4.79866629386968976635113149055, −4.07003732356991643742215120633, −3.75945380851443288105479501571, −3.32386278176966757284806766634, −2.62449455096830253760529803520, −2.35250834207176935545862591883, −1.89723013492434565879278243499, −1.36361238225807124477160395850, −0.17716958393716698301723713492,
0.17716958393716698301723713492, 1.36361238225807124477160395850, 1.89723013492434565879278243499, 2.35250834207176935545862591883, 2.62449455096830253760529803520, 3.32386278176966757284806766634, 3.75945380851443288105479501571, 4.07003732356991643742215120633, 4.79866629386968976635113149055, 5.36656148623248625541468556729, 5.37159320041324736889636203218, 5.81127752381652632262207622876, 5.92227696453935993708262691378, 6.85043000871931815896342313727, 6.92142553564184139105886482795, 7.45180014643452088743916269681, 7.84185637652786115042585901373, 7.925169463819268935943827772335, 8.690765611090871338018468546614, 8.895136151705939469094267602375
