| L(s) = 1 | − 2-s + 4-s − 4·5-s − 8-s + 4·10-s + 16-s − 4·20-s + 4·23-s + 6·25-s − 12·31-s − 32-s − 12·37-s + 4·40-s + 10·41-s − 12·43-s − 4·46-s − 12·49-s − 6·50-s − 4·59-s − 12·61-s + 12·62-s + 64-s + 8·73-s + 12·74-s − 4·80-s − 9·81-s − 10·82-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1.26·10-s + 1/4·16-s − 0.894·20-s + 0.834·23-s + 6/5·25-s − 2.15·31-s − 0.176·32-s − 1.97·37-s + 0.632·40-s + 1.56·41-s − 1.82·43-s − 0.589·46-s − 1.71·49-s − 0.848·50-s − 0.520·59-s − 1.53·61-s + 1.52·62-s + 1/8·64-s + 0.936·73-s + 1.39·74-s − 0.447·80-s − 81-s − 1.10·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 41 | $C_2$ | \( 1 - 10 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−10.43632157560539786744409074891, −9.780838612198209152964389925265, −9.172304898400516184959851872464, −8.654909790610720219398457035542, −8.269708165134371634604067106482, −7.52708418628645782538265070817, −7.35687597169912170795091836480, −6.76553334427942970576423644179, −5.94254501643540234482940439203, −5.11448942077865762253920862263, −4.45620217259664375200929268514, −3.54405772033426746848339849806, −3.21652125900131041898485109553, −1.74731173616796825880683426933, 0,
1.74731173616796825880683426933, 3.21652125900131041898485109553, 3.54405772033426746848339849806, 4.45620217259664375200929268514, 5.11448942077865762253920862263, 5.94254501643540234482940439203, 6.76553334427942970576423644179, 7.35687597169912170795091836480, 7.52708418628645782538265070817, 8.269708165134371634604067106482, 8.654909790610720219398457035542, 9.172304898400516184959851872464, 9.780838612198209152964389925265, 10.43632157560539786744409074891
