| L(s) = 1 | − 2·3-s + 4-s + 2·7-s + 9-s − 2·12-s − 3·16-s − 4·21-s − 2·25-s + 4·27-s + 2·28-s + 6·31-s + 36-s + 2·37-s + 6·43-s + 6·48-s − 2·49-s − 12·61-s + 2·63-s − 7·64-s − 4·67-s − 3·73-s + 4·75-s − 24·79-s − 11·81-s − 4·84-s − 12·93-s + 12·97-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 3/4·16-s − 0.872·21-s − 2/5·25-s + 0.769·27-s + 0.377·28-s + 1.07·31-s + 1/6·36-s + 0.328·37-s + 0.914·43-s + 0.866·48-s − 2/7·49-s − 1.53·61-s + 0.251·63-s − 7/8·64-s − 0.488·67-s − 0.351·73-s + 0.461·75-s − 2.70·79-s − 1.22·81-s − 0.436·84-s − 1.24·93-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 899433 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 899433 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.81657508022374816142574154032, −7.47974244213851709071765959422, −7.02486142002624652649609228216, −6.56809760205957940698926762757, −6.01803264149084837283616150989, −5.92388547056285945636848369886, −5.26758618357359319737677967021, −4.70400752712840579688722698551, −4.50542079225473482184512633050, −3.90385204149309413299766147551, −2.96844029189668345635428832359, −2.60491970689654138345620525509, −1.77304861700432957896836549928, −1.12780528890385860508149179679, 0,
1.12780528890385860508149179679, 1.77304861700432957896836549928, 2.60491970689654138345620525509, 2.96844029189668345635428832359, 3.90385204149309413299766147551, 4.50542079225473482184512633050, 4.70400752712840579688722698551, 5.26758618357359319737677967021, 5.92388547056285945636848369886, 6.01803264149084837283616150989, 6.56809760205957940698926762757, 7.02486142002624652649609228216, 7.47974244213851709071765959422, 7.81657508022374816142574154032
