| L(s) = 1 | + 2·3-s + 4·5-s + 9-s − 2·11-s + 8·15-s − 6·23-s + 11·25-s − 4·27-s + 8·31-s − 4·33-s + 10·37-s + 4·45-s + 14·47-s − 8·49-s + 10·53-s − 8·55-s − 8·59-s − 6·67-s − 12·69-s + 22·75-s − 11·81-s − 8·89-s + 16·93-s − 2·97-s − 2·99-s + 22·103-s + 20·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 1/3·9-s − 0.603·11-s + 2.06·15-s − 1.25·23-s + 11/5·25-s − 0.769·27-s + 1.43·31-s − 0.696·33-s + 1.64·37-s + 0.596·45-s + 2.04·47-s − 8/7·49-s + 1.37·53-s − 1.07·55-s − 1.04·59-s − 0.733·67-s − 1.44·69-s + 2.54·75-s − 1.22·81-s − 0.847·89-s + 1.65·93-s − 0.203·97-s − 0.201·99-s + 2.16·103-s + 1.89·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.796929978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.796929978\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.924257655855120773305509929217, −7.42602765568931877048731087278, −7.12610748186048586045884501761, −6.30431627502033767043419320477, −6.13022401632081364784206307219, −5.80218627605285542021994564539, −5.31578539968126551630699352938, −4.68888908475451977275407249765, −4.31232447967445935611600013234, −3.68760949928064187239591030843, −2.87281288029498066309348348619, −2.75390450274884111166400695119, −2.16597278581643182553002586458, −1.76083348668812613572221562979, −0.845344039240915771652762144061,
0.845344039240915771652762144061, 1.76083348668812613572221562979, 2.16597278581643182553002586458, 2.75390450274884111166400695119, 2.87281288029498066309348348619, 3.68760949928064187239591030843, 4.31232447967445935611600013234, 4.68888908475451977275407249765, 5.31578539968126551630699352938, 5.80218627605285542021994564539, 6.13022401632081364784206307219, 6.30431627502033767043419320477, 7.12610748186048586045884501761, 7.42602765568931877048731087278, 7.924257655855120773305509929217
