| L(s) = 1 | + 3·5-s + 7-s − 5·11-s + 4·13-s + 2·17-s + 6·19-s − 2·23-s + 5·25-s − 2·29-s + 31-s + 3·35-s − 10·37-s − 8·41-s + 8·43-s + 8·47-s − 6·49-s + 5·53-s − 15·55-s − 13·59-s + 8·61-s + 12·65-s + 14·67-s − 24·71-s + 6·73-s − 5·77-s − 11·79-s − 14·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.377·7-s − 1.50·11-s + 1.10·13-s + 0.485·17-s + 1.37·19-s − 0.417·23-s + 25-s − 0.371·29-s + 0.179·31-s + 0.507·35-s − 1.64·37-s − 1.24·41-s + 1.21·43-s + 1.16·47-s − 6/7·49-s + 0.686·53-s − 2.02·55-s − 1.69·59-s + 1.02·61-s + 1.48·65-s + 1.71·67-s − 2.84·71-s + 0.702·73-s − 0.569·77-s − 1.23·79-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.234569489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.234569489\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2} \) |
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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
−9.525774627127339485037275891585, −8.736691780547568924835719545975, −8.685204516555070348500925097758, −8.346747057119739796582791129875, −7.65793701233622131700668379039, −7.38534481682280194667432039612, −7.20361048881376163126012316129, −6.43638663487877139478220909891, −6.07935480851325592963968882750, −5.67423549159683224908794303725, −5.46896932051028645702667004199, −5.03700057765526876492813317456, −4.66948933101685461394439327788, −3.93798226426328505923379919355, −3.40943219604235664577552860876, −2.99938453640640460508747082895, −2.50033588674012115380288386634, −1.80777817784617706108728586105, −1.51261375670747518773645512088, −0.64836441377961573383700110765,
0.64836441377961573383700110765, 1.51261375670747518773645512088, 1.80777817784617706108728586105, 2.50033588674012115380288386634, 2.99938453640640460508747082895, 3.40943219604235664577552860876, 3.93798226426328505923379919355, 4.66948933101685461394439327788, 5.03700057765526876492813317456, 5.46896932051028645702667004199, 5.67423549159683224908794303725, 6.07935480851325592963968882750, 6.43638663487877139478220909891, 7.20361048881376163126012316129, 7.38534481682280194667432039612, 7.65793701233622131700668379039, 8.346747057119739796582791129875, 8.685204516555070348500925097758, 8.736691780547568924835719545975, 9.525774627127339485037275891585
