| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s + 4·13-s + 16-s + 6·22-s − 3·23-s − 4·25-s − 4·26-s − 32-s + 7·37-s − 6·44-s + 3·46-s + 3·47-s − 4·49-s + 4·50-s + 4·52-s + 6·59-s − 17·61-s + 64-s − 9·71-s − 2·73-s − 7·74-s + 3·83-s + 6·88-s − 3·92-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 1.10·13-s + 1/4·16-s + 1.27·22-s − 0.625·23-s − 4/5·25-s − 0.784·26-s − 0.176·32-s + 1.15·37-s − 0.904·44-s + 0.442·46-s + 0.437·47-s − 4/7·49-s + 0.565·50-s + 0.554·52-s + 0.781·59-s − 2.17·61-s + 1/8·64-s − 1.06·71-s − 0.234·73-s − 0.813·74-s + 0.329·83-s + 0.639·88-s − 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.754313887860678585642650051942, −8.267847849839037571968174676048, −7.916975231556401887365348684940, −7.62418836369938413605395359390, −7.05429072900205640367199706026, −6.32838155909901323672634845899, −5.94195824079011379761831550892, −5.54860452776490384871384529180, −4.85099677725082134639423748388, −4.21690158411758781870218999027, −3.51856134792203120441725440813, −2.82005214973772042781175894824, −2.25577800181923052862308336290, −1.32996351238838463542713950058, 0,
1.32996351238838463542713950058, 2.25577800181923052862308336290, 2.82005214973772042781175894824, 3.51856134792203120441725440813, 4.21690158411758781870218999027, 4.85099677725082134639423748388, 5.54860452776490384871384529180, 5.94195824079011379761831550892, 6.32838155909901323672634845899, 7.05429072900205640367199706026, 7.62418836369938413605395359390, 7.916975231556401887365348684940, 8.267847849839037571968174676048, 8.754313887860678585642650051942
