| L(s) = 1 | + 3-s + 9-s + 2·11-s − 4·13-s − 6·25-s + 27-s + 2·33-s + 20·37-s − 4·39-s − 16·47-s − 10·49-s + 24·59-s + 20·61-s + 16·71-s + 12·73-s − 6·75-s + 81-s + 32·83-s − 4·97-s + 2·99-s − 20·109-s + 20·111-s − 4·117-s + 3·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 6/5·25-s + 0.192·27-s + 0.348·33-s + 3.28·37-s − 0.640·39-s − 2.33·47-s − 1.42·49-s + 3.12·59-s + 2.56·61-s + 1.89·71-s + 1.40·73-s − 0.692·75-s + 1/9·81-s + 3.51·83-s − 0.406·97-s + 0.201·99-s − 1.91·109-s + 1.89·111-s − 0.369·117-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.036916169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036916169\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−9.413820553443583654003786378045, −8.352307473563578559330404703156, −7.996057650348538062805293058353, −7.995283121669507299025547417770, −7.08919828768618765963206395693, −6.70453643193197308588262041684, −6.31116345632917881418704272529, −5.57273521417705379772496968279, −5.01052020020146862034178505578, −4.52635368073741687498225693994, −3.79294235641559942977546839028, −3.47375416333532803042906659795, −2.35991906031001647976266238936, −2.21252199435586523776406167130, −0.890313293886189684730186743974,
0.890313293886189684730186743974, 2.21252199435586523776406167130, 2.35991906031001647976266238936, 3.47375416333532803042906659795, 3.79294235641559942977546839028, 4.52635368073741687498225693994, 5.01052020020146862034178505578, 5.57273521417705379772496968279, 6.31116345632917881418704272529, 6.70453643193197308588262041684, 7.08919828768618765963206395693, 7.995283121669507299025547417770, 7.996057650348538062805293058353, 8.352307473563578559330404703156, 9.413820553443583654003786378045
