| L(s) = 1 | − 3·5-s + 9-s + 2·11-s − 4·16-s − 2·19-s + 4·25-s − 4·29-s − 12·31-s − 3·45-s + 11·49-s − 6·55-s − 16·59-s − 2·61-s − 24·71-s + 32·79-s + 12·80-s + 81-s − 12·89-s + 6·95-s + 2·99-s + 4·101-s + 8·109-s − 19·121-s + 3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1/3·9-s + 0.603·11-s − 16-s − 0.458·19-s + 4/5·25-s − 0.742·29-s − 2.15·31-s − 0.447·45-s + 11/7·49-s − 0.809·55-s − 2.08·59-s − 0.256·61-s − 2.84·71-s + 3.60·79-s + 1.34·80-s + 1/9·81-s − 1.27·89-s + 0.615·95-s + 0.201·99-s + 0.398·101-s + 0.766·109-s − 1.72·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−9.341053920747992022463934553266, −8.836427381447004052087018681826, −8.723409361965311529712840067335, −7.74191106110524052069368926798, −7.47848778675296589128891353641, −7.15886697828319609358757774448, −6.41173993136661837405396202601, −5.94366419557132992161838079855, −5.07072622007111059188732569210, −4.55783516638495121923212657411, −3.81591641511052084615077615666, −3.69330683704517818549980694450, −2.57494569661476323183620903482, −1.60475124369086001622436622243, 0,
1.60475124369086001622436622243, 2.57494569661476323183620903482, 3.69330683704517818549980694450, 3.81591641511052084615077615666, 4.55783516638495121923212657411, 5.07072622007111059188732569210, 5.94366419557132992161838079855, 6.41173993136661837405396202601, 7.15886697828319609358757774448, 7.47848778675296589128891353641, 7.74191106110524052069368926798, 8.723409361965311529712840067335, 8.836427381447004052087018681826, 9.341053920747992022463934553266
