| L(s) = 1 | − 3-s − 4-s + 9-s + 12-s + 8·13-s − 3·16-s − 10·25-s − 27-s + 2·31-s − 36-s − 8·39-s + 4·43-s + 3·48-s − 10·49-s − 8·52-s + 5·61-s + 7·64-s − 14·67-s + 12·73-s + 10·75-s − 24·79-s + 81-s − 2·93-s + 10·100-s + 6·103-s + 108-s − 8·109-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1/3·9-s + 0.288·12-s + 2.21·13-s − 3/4·16-s − 2·25-s − 0.192·27-s + 0.359·31-s − 1/6·36-s − 1.28·39-s + 0.609·43-s + 0.433·48-s − 1.42·49-s − 1.10·52-s + 0.640·61-s + 7/8·64-s − 1.71·67-s + 1.40·73-s + 1.15·75-s − 2.70·79-s + 1/9·81-s − 0.207·93-s + 100-s + 0.591·103-s + 0.0962·108-s − 0.766·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574803 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574803 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| 349 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 30 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.281217586751022569677766440865, −7.73959378953191402198924106991, −7.41374644686749385323244056043, −6.62918127711846836746394172878, −6.34065692077851685742247786286, −5.94552161770747711159968367770, −5.53250184867588968167216982818, −4.92336044520202239140924116987, −4.33629138464565658429913630646, −3.91665898547505411407587946160, −3.55170975021472970196854244915, −2.69916628897736680521078117496, −1.81261135717038625316232999552, −1.17979705882738141498221009990, 0,
1.17979705882738141498221009990, 1.81261135717038625316232999552, 2.69916628897736680521078117496, 3.55170975021472970196854244915, 3.91665898547505411407587946160, 4.33629138464565658429913630646, 4.92336044520202239140924116987, 5.53250184867588968167216982818, 5.94552161770747711159968367770, 6.34065692077851685742247786286, 6.62918127711846836746394172878, 7.41374644686749385323244056043, 7.73959378953191402198924106991, 8.281217586751022569677766440865
