| L(s) = 1 | − 4·5-s − 3·9-s − 6·11-s − 16·19-s + 11·25-s + 2·29-s + 4·31-s − 12·41-s + 12·45-s − 49-s + 24·55-s − 20·59-s − 14·79-s − 16·89-s + 64·95-s + 18·99-s − 24·101-s + 14·109-s + 5·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 9-s − 1.80·11-s − 3.67·19-s + 11/5·25-s + 0.371·29-s + 0.718·31-s − 1.87·41-s + 1.78·45-s − 1/7·49-s + 3.23·55-s − 2.60·59-s − 1.57·79-s − 1.69·89-s + 6.56·95-s + 1.80·99-s − 2.38·101-s + 1.34·109-s + 5/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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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
−10.58804003912029222761714093689, −10.39122221674609264460421941462, −9.813340107754439583591978097041, −8.835539964443643956614724393747, −8.673555457906657883500618249836, −8.199690345447704683467608904797, −8.129436152953682363906960379319, −7.58129235973462725917552446622, −6.90625967382598233767942439212, −6.54283079933294675299992408879, −6.00644602808738444314514610434, −5.36047721508849218345547444582, −4.72136855038995652583907868073, −4.40118322555487914118174305000, −3.91813131858336118609679447058, −2.90182251378309740686117137120, −2.89313021270953584683833638981, −1.90029477991754274331086169687, 0, 0,
1.90029477991754274331086169687, 2.89313021270953584683833638981, 2.90182251378309740686117137120, 3.91813131858336118609679447058, 4.40118322555487914118174305000, 4.72136855038995652583907868073, 5.36047721508849218345547444582, 6.00644602808738444314514610434, 6.54283079933294675299992408879, 6.90625967382598233767942439212, 7.58129235973462725917552446622, 8.129436152953682363906960379319, 8.199690345447704683467608904797, 8.673555457906657883500618249836, 8.835539964443643956614724393747, 9.813340107754439583591978097041, 10.39122221674609264460421941462, 10.58804003912029222761714093689
