| L(s) = 1 | − 3-s − 4-s + 9-s + 12-s + 2·13-s + 16-s + 19-s + 6·25-s − 27-s + 2·31-s − 36-s + 6·37-s − 2·39-s − 8·43-s − 48-s − 10·49-s − 2·52-s − 57-s − 12·61-s − 64-s − 16·67-s − 16·73-s − 6·75-s − 76-s + 2·79-s + 81-s − 2·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.229·19-s + 6/5·25-s − 0.192·27-s + 0.359·31-s − 1/6·36-s + 0.986·37-s − 0.320·39-s − 1.21·43-s − 0.144·48-s − 1.42·49-s − 0.277·52-s − 0.132·57-s − 1.53·61-s − 1/8·64-s − 1.95·67-s − 1.87·73-s − 0.692·75-s − 0.114·76-s + 0.225·79-s + 1/9·81-s − 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.183543219577224388436904152477, −7.63630335589458345937232249879, −7.14083961353630804979395184532, −6.63476570475728421128822946853, −6.28451854901339919449557531977, −5.77763744585875121435967355110, −5.34110224567434626668185813894, −4.74329972392802860888192543540, −4.46245327973518306899198622211, −3.92046810864675229498181414898, −3.07364753727346617579683991524, −2.90060183359140465290290245010, −1.68407289406377645559105544028, −1.15856078640089366274780288036, 0,
1.15856078640089366274780288036, 1.68407289406377645559105544028, 2.90060183359140465290290245010, 3.07364753727346617579683991524, 3.92046810864675229498181414898, 4.46245327973518306899198622211, 4.74329972392802860888192543540, 5.34110224567434626668185813894, 5.77763744585875121435967355110, 6.28451854901339919449557531977, 6.63476570475728421128822946853, 7.14083961353630804979395184532, 7.63630335589458345937232249879, 8.183543219577224388436904152477
