| L(s) = 1 | − 2-s − 7-s + 8-s + 9-s + 14-s − 16-s + 3·17-s − 18-s − 23-s + 3·31-s − 3·34-s + 46-s + 3·47-s − 56-s − 3·62-s − 63-s + 64-s + 2·71-s + 72-s − 79-s + 3·89-s − 3·94-s − 3·103-s + 112-s − 2·113-s − 3·119-s − 121-s + ⋯ |
| L(s) = 1 | − 2-s − 7-s + 8-s + 9-s + 14-s − 16-s + 3·17-s − 18-s − 23-s + 3·31-s − 3·34-s + 46-s + 3·47-s − 56-s − 3·62-s − 63-s + 64-s + 2·71-s + 72-s − 79-s + 3·89-s − 3·94-s − 3·103-s + 112-s − 2·113-s − 3·119-s − 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6839190028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6839190028\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.03356353414453699103718349205, −9.644215447676611310771336950373, −9.228480689166928959682455663129, −8.916920055650879232363472994235, −8.156488497678090914975982233093, −8.051645803199376206093487782013, −7.53180114653832798300357303879, −7.50301211291397246961275576037, −6.57608681935525486481620078292, −6.54981025084636754142039533373, −5.94927884281690563030827749250, −5.27167970366015021865604844389, −5.15349239292167639414577953861, −4.23784975096476487722017851375, −4.00983701707170394747777622196, −3.54415359655160071793357103862, −2.80887601638612593341476237134, −2.36846604699452952415481155773, −1.14555718080571493573882026365, −1.11729674022285314441964022612,
1.11729674022285314441964022612, 1.14555718080571493573882026365, 2.36846604699452952415481155773, 2.80887601638612593341476237134, 3.54415359655160071793357103862, 4.00983701707170394747777622196, 4.23784975096476487722017851375, 5.15349239292167639414577953861, 5.27167970366015021865604844389, 5.94927884281690563030827749250, 6.54981025084636754142039533373, 6.57608681935525486481620078292, 7.50301211291397246961275576037, 7.53180114653832798300357303879, 8.051645803199376206093487782013, 8.156488497678090914975982233093, 8.916920055650879232363472994235, 9.228480689166928959682455663129, 9.644215447676611310771336950373, 10.03356353414453699103718349205
