| L(s) = 1 | + 4-s + 6·11-s + 16·19-s + 12·29-s − 16·31-s − 12·41-s + 6·44-s + 2·49-s + 18·59-s − 16·61-s − 64-s + 24·71-s + 16·76-s + 16·79-s − 36·89-s + 24·101-s − 8·109-s + 12·116-s + 31·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.80·11-s + 3.67·19-s + 2.22·29-s − 2.87·31-s − 1.87·41-s + 0.904·44-s + 2/7·49-s + 2.34·59-s − 2.04·61-s − 1/8·64-s + 2.84·71-s + 1.83·76-s + 1.80·79-s − 3.81·89-s + 2.38·101-s − 0.766·109-s + 1.11·116-s + 2.81·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.077026820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.077026820\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 50 T^{2} + 291 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 85 T^{2} + 336 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 145 T^{2} + 11616 T^{4} + 145 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.96103248341772791580574854932, −6.68762952543781297888482717959, −6.53627746934021715804196715200, −6.20624551650071824098263319084, −6.05620043670249728664981945534, −5.65277794167311098156360111720, −5.42576730818787627611817443855, −5.38562547495512621390246873111, −5.30818309224393171443392242219, −4.82638874836616435354094745209, −4.64763689979305352590936062994, −4.30178181156430416013534151535, −4.24731318395897550072025837613, −3.69406752965120113980033216685, −3.44860247441931919091062349752, −3.36099241622914863781522462628, −3.35276196464029918220220500543, −2.96298453664751836794213367225, −2.54135215374031022065314848866, −2.10200841320752724310977369664, −1.99921557173778390779102155183, −1.54560008975365690710728840645, −1.09660241641646530651130576284, −1.09592325120560405604216929971, −0.52001905300213681786678967794,
0.52001905300213681786678967794, 1.09592325120560405604216929971, 1.09660241641646530651130576284, 1.54560008975365690710728840645, 1.99921557173778390779102155183, 2.10200841320752724310977369664, 2.54135215374031022065314848866, 2.96298453664751836794213367225, 3.35276196464029918220220500543, 3.36099241622914863781522462628, 3.44860247441931919091062349752, 3.69406752965120113980033216685, 4.24731318395897550072025837613, 4.30178181156430416013534151535, 4.64763689979305352590936062994, 4.82638874836616435354094745209, 5.30818309224393171443392242219, 5.38562547495512621390246873111, 5.42576730818787627611817443855, 5.65277794167311098156360111720, 6.05620043670249728664981945534, 6.20624551650071824098263319084, 6.53627746934021715804196715200, 6.68762952543781297888482717959, 6.96103248341772791580574854932
