| L(s) = 1 | + 4-s + 2·7-s + 4·13-s + 16-s − 14·19-s − 25-s + 2·28-s + 4·31-s + 4·37-s + 4·43-s + 3·49-s + 4·52-s + 10·61-s + 64-s + 16·67-s + 4·73-s − 14·76-s + 10·79-s + 8·91-s + 4·97-s − 100-s − 20·103-s − 20·109-s + 2·112-s + 14·121-s + 4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 1.10·13-s + 1/4·16-s − 3.21·19-s − 1/5·25-s + 0.377·28-s + 0.718·31-s + 0.657·37-s + 0.609·43-s + 3/7·49-s + 0.554·52-s + 1.28·61-s + 1/8·64-s + 1.95·67-s + 0.468·73-s − 1.60·76-s + 1.12·79-s + 0.838·91-s + 0.406·97-s − 0.0999·100-s − 1.97·103-s − 1.91·109-s + 0.188·112-s + 1.27·121-s + 0.359·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.576348256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576348256\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.094785783071409988385687751137, −7.73532298603754652567815778453, −6.84669532198488939637919288494, −6.73871282898076223759796265441, −6.36293944731828035024321117773, −5.73900595462088482953892956618, −5.56957044429086118267228114847, −4.59790954336953877055233514712, −4.49364756499371052867692865672, −3.87247358034755873056669071066, −3.48588234639422265078311386338, −2.44868872871059860732796494913, −2.28395108895772497375168075701, −1.60097986545961424167062353389, −0.71201387828374154761208002119,
0.71201387828374154761208002119, 1.60097986545961424167062353389, 2.28395108895772497375168075701, 2.44868872871059860732796494913, 3.48588234639422265078311386338, 3.87247358034755873056669071066, 4.49364756499371052867692865672, 4.59790954336953877055233514712, 5.56957044429086118267228114847, 5.73900595462088482953892956618, 6.36293944731828035024321117773, 6.73871282898076223759796265441, 6.84669532198488939637919288494, 7.73532298603754652567815778453, 8.094785783071409988385687751137
