| L(s) = 1 | − 2·3-s − 3·4-s − 4·5-s − 3·9-s + 3·11-s + 6·12-s + 8·15-s + 5·16-s + 12·20-s − 8·23-s + 2·25-s + 14·27-s + 10·31-s − 6·33-s + 9·36-s − 22·37-s − 9·44-s + 12·45-s − 10·48-s − 5·49-s + 12·53-s − 12·55-s + 10·59-s − 24·60-s − 3·64-s − 4·67-s + 16·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s − 1.78·5-s − 9-s + 0.904·11-s + 1.73·12-s + 2.06·15-s + 5/4·16-s + 2.68·20-s − 1.66·23-s + 2/5·25-s + 2.69·27-s + 1.79·31-s − 1.04·33-s + 3/2·36-s − 3.61·37-s − 1.35·44-s + 1.78·45-s − 1.44·48-s − 5/7·49-s + 1.64·53-s − 1.61·55-s + 1.30·59-s − 3.09·60-s − 3/8·64-s − 0.488·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.118927993931557797577222055683, −7.24884684342390107531743170326, −6.97450299304430271706246705192, −6.24077886297520789663122777596, −6.00831015099480609083754328355, −5.31327024180315445602831381888, −5.10251401460890303589286087658, −4.49871079624823310774196588431, −4.05031920828601961060326774393, −3.65475901244453246589839654026, −3.31355572200378673558994513483, −2.27615150194359396668418676133, −1.02383200139573404740051201822, 0, 0,
1.02383200139573404740051201822, 2.27615150194359396668418676133, 3.31355572200378673558994513483, 3.65475901244453246589839654026, 4.05031920828601961060326774393, 4.49871079624823310774196588431, 5.10251401460890303589286087658, 5.31327024180315445602831381888, 6.00831015099480609083754328355, 6.24077886297520789663122777596, 6.97450299304430271706246705192, 7.24884684342390107531743170326, 8.118927993931557797577222055683
