| L(s) = 1 | − 3·5-s − 7-s − 5·11-s + 4·13-s − 2·17-s − 6·19-s − 2·23-s + 5·25-s + 2·29-s − 31-s + 3·35-s − 10·37-s + 8·41-s − 8·43-s + 8·47-s − 6·49-s − 5·53-s + 15·55-s − 13·59-s + 8·61-s − 12·65-s − 14·67-s − 24·71-s + 6·73-s + 5·77-s + 11·79-s − 14·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 1.50·11-s + 1.10·13-s − 0.485·17-s − 1.37·19-s − 0.417·23-s + 25-s + 0.371·29-s − 0.179·31-s + 0.507·35-s − 1.64·37-s + 1.24·41-s − 1.21·43-s + 1.16·47-s − 6/7·49-s − 0.686·53-s + 2.02·55-s − 1.69·59-s + 1.02·61-s − 1.48·65-s − 1.71·67-s − 2.84·71-s + 0.702·73-s + 0.569·77-s + 1.23·79-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2} \) |
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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
−8.749867596504237817483395800735, −8.574163700162763437460490549170, −8.256345943239460385592664359877, −7.75423828573436017207993193889, −7.35467207236812755498711981314, −7.30554018304380329206207706917, −6.36039376761233964070766062046, −6.33454626934652650177095189673, −5.93906035099465559644887012914, −5.18855358271030214463229728234, −4.88738454659966614353772842037, −4.38590756944811627051974340652, −4.04621587594847349989881514061, −3.45832543906301199020577419066, −3.19999640386604099858669326856, −2.56480927008343591572797618993, −2.01417978577259598031825874161, −1.24010562791137743185478618603, 0, 0,
1.24010562791137743185478618603, 2.01417978577259598031825874161, 2.56480927008343591572797618993, 3.19999640386604099858669326856, 3.45832543906301199020577419066, 4.04621587594847349989881514061, 4.38590756944811627051974340652, 4.88738454659966614353772842037, 5.18855358271030214463229728234, 5.93906035099465559644887012914, 6.33454626934652650177095189673, 6.36039376761233964070766062046, 7.30554018304380329206207706917, 7.35467207236812755498711981314, 7.75423828573436017207993193889, 8.256345943239460385592664359877, 8.574163700162763437460490549170, 8.749867596504237817483395800735
