L(s) = 1 | − 9-s − 2·11-s − 6·19-s − 12·29-s − 8·31-s + 4·41-s − 6·49-s + 4·59-s − 8·61-s + 14·71-s + 2·79-s + 81-s − 20·89-s + 2·99-s + 8·101-s − 16·109-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + ⋯ |
L(s) = 1 | − 1/3·9-s − 0.603·11-s − 1.37·19-s − 2.22·29-s − 1.43·31-s + 0.624·41-s − 6/7·49-s + 0.520·59-s − 1.02·61-s + 1.66·71-s + 0.225·79-s + 1/9·81-s − 2.11·89-s + 0.201·99-s + 0.796·101-s − 1.53·109-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
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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
−9.422197574188130817763059589262, −8.965053913275388148374816983982, −8.356973850486090935420059188355, −7.957883933574345054701003107581, −7.41998954855005023127010946195, −6.91314094544873394457774872417, −6.30863413441593372512421985066, −5.63139352270634351168527672090, −5.39543344743137463832307303895, −4.55454017653786936444030370390, −3.93635944741692071534317197982, −3.34241924400158333321788587678, −2.41986240838994355096835391492, −1.77673945351887851385720722094, 0,
1.77673945351887851385720722094, 2.41986240838994355096835391492, 3.34241924400158333321788587678, 3.93635944741692071534317197982, 4.55454017653786936444030370390, 5.39543344743137463832307303895, 5.63139352270634351168527672090, 6.30863413441593372512421985066, 6.91314094544873394457774872417, 7.41998954855005023127010946195, 7.957883933574345054701003107581, 8.356973850486090935420059188355, 8.965053913275388148374816983982, 9.422197574188130817763059589262