| L(s) = 1 | − 4-s − 4·5-s + 8·11-s + 16-s + 2·19-s + 4·20-s + 11·25-s − 4·29-s + 24·41-s − 8·44-s + 14·49-s − 32·55-s − 20·59-s − 12·61-s − 64-s − 24·71-s − 2·76-s + 16·79-s − 4·80-s − 16·89-s − 8·95-s − 11·100-s − 24·101-s + 28·109-s + 4·116-s + 26·121-s − 24·125-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s + 2.41·11-s + 1/4·16-s + 0.458·19-s + 0.894·20-s + 11/5·25-s − 0.742·29-s + 3.74·41-s − 1.20·44-s + 2·49-s − 4.31·55-s − 2.60·59-s − 1.53·61-s − 1/8·64-s − 2.84·71-s − 0.229·76-s + 1.80·79-s − 0.447·80-s − 1.69·89-s − 0.820·95-s − 1.09·100-s − 2.38·101-s + 2.68·109-s + 0.371·116-s + 2.36·121-s − 2.14·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.576612225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576612225\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.374370911423795205274291342450, −9.292024342077516206329770052315, −8.647348584629239805046654344710, −8.535367432913759626836397428952, −7.83842740388649712558927411850, −7.46071392701930798830936154448, −7.36732610777691659276556120731, −6.88956734081434786106101238285, −6.30341858174313776356701208912, −5.85263583998635103419103169491, −5.66712805410295926267963157100, −4.58029228141831859332207397481, −4.50172493008255613495263289013, −4.08181108724219159152351296004, −3.87982933375635799366194039024, −3.13051187884498796533339256776, −2.94209847647220260044219177673, −1.81626712317735118204374675795, −1.13232696657335121694437892039, −0.58515639726989476720516646524,
0.58515639726989476720516646524, 1.13232696657335121694437892039, 1.81626712317735118204374675795, 2.94209847647220260044219177673, 3.13051187884498796533339256776, 3.87982933375635799366194039024, 4.08181108724219159152351296004, 4.50172493008255613495263289013, 4.58029228141831859332207397481, 5.66712805410295926267963157100, 5.85263583998635103419103169491, 6.30341858174313776356701208912, 6.88956734081434786106101238285, 7.36732610777691659276556120731, 7.46071392701930798830936154448, 7.83842740388649712558927411850, 8.535367432913759626836397428952, 8.647348584629239805046654344710, 9.292024342077516206329770052315, 9.374370911423795205274291342450
