| L(s) = 1 | − 2·3-s + 4-s + 5-s + 7-s + 9-s − 2·12-s − 2·15-s − 3·16-s − 2·17-s + 20-s − 2·21-s + 25-s + 4·27-s + 28-s + 35-s + 36-s − 2·37-s − 12·41-s + 4·43-s + 45-s − 8·47-s + 6·48-s + 49-s + 4·51-s + 4·59-s − 2·60-s + 63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.516·15-s − 3/4·16-s − 0.485·17-s + 0.223·20-s − 0.436·21-s + 1/5·25-s + 0.769·27-s + 0.188·28-s + 0.169·35-s + 1/6·36-s − 0.328·37-s − 1.87·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 0.866·48-s + 1/7·49-s + 0.560·51-s + 0.520·59-s − 0.258·60-s + 0.125·63-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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
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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
−8.368278137903934811161180143152, −8.105207974465289144440035940584, −7.28573009705324113320809721046, −6.86368073595582829586047849016, −6.60380143707028668939869882005, −6.17342899499695696728054179407, −5.52455487993512115102528639141, −5.21521895070743645987729037953, −4.73903914031453691952522686918, −4.17790225011024193050996467821, −3.41422597748180196055783944094, −2.65092800876268994123133811746, −2.04368680713134052582765914207, −1.27711585002054323917962832736, 0,
1.27711585002054323917962832736, 2.04368680713134052582765914207, 2.65092800876268994123133811746, 3.41422597748180196055783944094, 4.17790225011024193050996467821, 4.73903914031453691952522686918, 5.21521895070743645987729037953, 5.52455487993512115102528639141, 6.17342899499695696728054179407, 6.60380143707028668939869882005, 6.86368073595582829586047849016, 7.28573009705324113320809721046, 8.105207974465289144440035940584, 8.368278137903934811161180143152
