| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s + 15-s − 4·16-s − 2·17-s − 21-s + 25-s + 5·27-s − 35-s + 16·43-s + 2·45-s − 6·47-s + 4·48-s + 49-s + 2·51-s − 2·63-s + 16·67-s − 75-s + 2·79-s + 4·80-s + 81-s + 8·83-s + 2·85-s − 16·89-s − 16·101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.258·15-s − 16-s − 0.485·17-s − 0.218·21-s + 1/5·25-s + 0.962·27-s − 0.169·35-s + 2.43·43-s + 0.298·45-s − 0.875·47-s + 0.577·48-s + 1/7·49-s + 0.280·51-s − 0.251·63-s + 1.95·67-s − 0.115·75-s + 0.225·79-s + 0.447·80-s + 1/9·81-s + 0.878·83-s + 0.216·85-s − 1.69·89-s − 1.59·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
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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
−8.341102353525611125768436023206, −8.063990577145466610163076570908, −7.55657033798281309555617548880, −6.92694677919713909549320793387, −6.63358719692764678873952587865, −6.13220268257666973731970391644, −5.49720704876876926398808862858, −5.18520141932526353514078079211, −4.52530893472467518374574956578, −4.16015078768670268188311759815, −3.50086264620923726472432189081, −2.64937413738378276694747792046, −2.26174033185137859717630141920, −1.10054144857796676546272074331, 0,
1.10054144857796676546272074331, 2.26174033185137859717630141920, 2.64937413738378276694747792046, 3.50086264620923726472432189081, 4.16015078768670268188311759815, 4.52530893472467518374574956578, 5.18520141932526353514078079211, 5.49720704876876926398808862858, 6.13220268257666973731970391644, 6.63358719692764678873952587865, 6.92694677919713909549320793387, 7.55657033798281309555617548880, 8.063990577145466610163076570908, 8.341102353525611125768436023206
