| L(s) = 1 | − 2-s + 3·5-s + 5·7-s + 8-s − 3·10-s + 3·11-s − 8·13-s − 5·14-s − 16-s + 4·19-s − 3·22-s + 5·25-s + 8·26-s − 18·29-s + 31-s + 15·35-s − 8·37-s − 4·38-s + 3·40-s − 20·43-s − 6·47-s + 18·49-s − 5·50-s − 3·53-s + 9·55-s + 5·56-s + 18·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·5-s + 1.88·7-s + 0.353·8-s − 0.948·10-s + 0.904·11-s − 2.21·13-s − 1.33·14-s − 1/4·16-s + 0.917·19-s − 0.639·22-s + 25-s + 1.56·26-s − 3.34·29-s + 0.179·31-s + 2.53·35-s − 1.31·37-s − 0.648·38-s + 0.474·40-s − 3.04·43-s − 0.875·47-s + 18/7·49-s − 0.707·50-s − 0.412·53-s + 1.21·55-s + 0.668·56-s + 2.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045258298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045258298\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−13.48823057964127948742975721049, −13.45617691146784018129781377503, −12.48342779363780871045839587045, −12.01316835565085929352361181823, −11.45061696522902661556017684510, −11.14092985233623642253676869890, −10.35271728185846852446407742427, −9.849485421302246263818797824799, −9.399569344645120450055788494808, −9.194333467888403441282220988661, −8.151213123067926269352967750145, −7.994407119689783532594940018411, −6.97549366422668717138505761545, −6.93794384574737958812683295302, −5.41390844351482264400565507817, −5.34835803104314458794293990465, −4.73939802009150107229071478196, −3.60582205044088538160706793792, −2.04685327077511662941445831714, −1.75386172048355320554229063194,
1.75386172048355320554229063194, 2.04685327077511662941445831714, 3.60582205044088538160706793792, 4.73939802009150107229071478196, 5.34835803104314458794293990465, 5.41390844351482264400565507817, 6.93794384574737958812683295302, 6.97549366422668717138505761545, 7.994407119689783532594940018411, 8.151213123067926269352967750145, 9.194333467888403441282220988661, 9.399569344645120450055788494808, 9.849485421302246263818797824799, 10.35271728185846852446407742427, 11.14092985233623642253676869890, 11.45061696522902661556017684510, 12.01316835565085929352361181823, 12.48342779363780871045839587045, 13.45617691146784018129781377503, 13.48823057964127948742975721049
