| L(s) = 1 | + 2·3-s − 2·7-s + 2·9-s − 2·13-s − 2·17-s + 8·19-s − 4·21-s + 10·23-s + 6·27-s − 10·37-s − 4·39-s + 12·41-s − 6·43-s + 14·47-s + 2·49-s − 4·51-s − 2·53-s + 16·57-s + 8·59-s − 4·61-s − 4·63-s + 14·67-s + 20·69-s − 18·73-s + 16·79-s + 11·81-s + 10·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 2/3·9-s − 0.554·13-s − 0.485·17-s + 1.83·19-s − 0.872·21-s + 2.08·23-s + 1.15·27-s − 1.64·37-s − 0.640·39-s + 1.87·41-s − 0.914·43-s + 2.04·47-s + 2/7·49-s − 0.560·51-s − 0.274·53-s + 2.11·57-s + 1.04·59-s − 0.512·61-s − 0.503·63-s + 1.71·67-s + 2.40·69-s − 2.10·73-s + 1.80·79-s + 11/9·81-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.541791827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.541791827\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.399138532315601929631861252584, −9.243072371392499420253651797293, −8.797798566465001675462334830021, −8.683378625186750792913895541591, −7.908526189653864853575777906300, −7.69670304685218610623931435606, −7.16307409618371448895878095344, −6.93727411485892768383615336976, −6.62869100103402877147323772212, −5.95393472857215121955830003603, −5.33786599831243196001657380320, −5.15604000074851863198251272913, −4.59793702833499293242457194665, −4.01837887188466131351765351014, −3.34403631815961901670637454084, −3.28253317423061125992855079629, −2.55911487843505104698930491476, −2.39312566231043210505475636148, −1.33689939001090117967721838301, −0.73496640570089924081057845686,
0.73496640570089924081057845686, 1.33689939001090117967721838301, 2.39312566231043210505475636148, 2.55911487843505104698930491476, 3.28253317423061125992855079629, 3.34403631815961901670637454084, 4.01837887188466131351765351014, 4.59793702833499293242457194665, 5.15604000074851863198251272913, 5.33786599831243196001657380320, 5.95393472857215121955830003603, 6.62869100103402877147323772212, 6.93727411485892768383615336976, 7.16307409618371448895878095344, 7.69670304685218610623931435606, 7.908526189653864853575777906300, 8.683378625186750792913895541591, 8.797798566465001675462334830021, 9.243072371392499420253651797293, 9.399138532315601929631861252584
