| L(s) = 1 | + 2·3-s − 4-s − 3·9-s − 2·12-s − 3·16-s − 14·27-s − 16·31-s + 3·36-s + 16·37-s − 18·47-s − 6·48-s − 11·49-s − 12·53-s − 24·59-s + 7·64-s + 10·67-s − 24·71-s − 4·81-s + 6·89-s − 32·93-s + 20·97-s + 8·103-s + 14·108-s + 32·111-s + 12·113-s + 16·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 9-s − 0.577·12-s − 3/4·16-s − 2.69·27-s − 2.87·31-s + 1/2·36-s + 2.63·37-s − 2.62·47-s − 0.866·48-s − 1.57·49-s − 1.64·53-s − 3.12·59-s + 7/8·64-s + 1.22·67-s − 2.84·71-s − 4/9·81-s + 0.635·89-s − 3.31·93-s + 2.03·97-s + 0.788·103-s + 1.34·108-s + 3.03·111-s + 1.12·113-s + 1.43·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.451893956446662929139941958392, −8.327171607554445491106677120366, −7.77380019299085524138115369225, −7.51653018145375545501561844481, −7.42085387940016343638920601121, −6.42690181856603608336809351508, −6.19345110913961570096630669202, −6.12227347335132055088873058701, −5.25911822446654566921719686456, −5.16639885545443726412690652954, −4.45329178128836373699649684300, −4.35312622036818861002172526793, −3.44420246050137511187095649367, −3.31542274072744033238996171940, −3.04598216573898879678433334136, −2.31968796637188150022728979119, −1.95915881132655300042000866907, −1.40362957628394311272625414820, 0, 0,
1.40362957628394311272625414820, 1.95915881132655300042000866907, 2.31968796637188150022728979119, 3.04598216573898879678433334136, 3.31542274072744033238996171940, 3.44420246050137511187095649367, 4.35312622036818861002172526793, 4.45329178128836373699649684300, 5.16639885545443726412690652954, 5.25911822446654566921719686456, 6.12227347335132055088873058701, 6.19345110913961570096630669202, 6.42690181856603608336809351508, 7.42085387940016343638920601121, 7.51653018145375545501561844481, 7.77380019299085524138115369225, 8.327171607554445491106677120366, 8.451893956446662929139941958392
