| L(s) = 1 | + 2·5-s + 4·13-s + 12·17-s + 3·25-s − 12·29-s + 4·37-s − 12·41-s − 10·49-s + 12·53-s + 4·61-s + 8·65-s + 4·73-s + 24·85-s + 12·89-s + 4·97-s − 12·101-s + 4·109-s + 12·113-s − 22·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.10·13-s + 2.91·17-s + 3/5·25-s − 2.22·29-s + 0.657·37-s − 1.87·41-s − 1.42·49-s + 1.64·53-s + 0.512·61-s + 0.992·65-s + 0.468·73-s + 2.60·85-s + 1.27·89-s + 0.406·97-s − 1.19·101-s + 0.383·109-s + 1.12·113-s − 2·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141031885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141031885\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−9.633261704500415779063377753491, −8.847850276232423443211930888771, −8.536040373596851277524077678909, −7.74818737522785607031159368119, −7.64589778281564359800589548721, −6.88656092970954199609923322632, −6.21444115782927123192608967686, −5.90138678053477774939394585997, −5.19647848534431193907436500640, −5.17606419420266207753384183426, −3.71119793411816721603957340078, −3.70610300459195515417889608627, −2.84705552122166969163585152299, −1.81793015252092636076156145980, −1.16915866227454488376692012454,
1.16915866227454488376692012454, 1.81793015252092636076156145980, 2.84705552122166969163585152299, 3.70610300459195515417889608627, 3.71119793411816721603957340078, 5.17606419420266207753384183426, 5.19647848534431193907436500640, 5.90138678053477774939394585997, 6.21444115782927123192608967686, 6.88656092970954199609923322632, 7.64589778281564359800589548721, 7.74818737522785607031159368119, 8.536040373596851277524077678909, 8.847850276232423443211930888771, 9.633261704500415779063377753491
