| L(s) = 1 | + 3-s + 9-s − 4·13-s + 8·23-s + 2·25-s + 27-s + 4·37-s − 4·39-s + 8·47-s + 2·49-s + 24·59-s + 4·61-s + 8·69-s + 8·71-s − 12·73-s + 2·75-s + 81-s − 16·83-s − 12·97-s + 8·107-s + 12·109-s + 4·111-s − 4·117-s − 6·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.10·13-s + 1.66·23-s + 2/5·25-s + 0.192·27-s + 0.657·37-s − 0.640·39-s + 1.16·47-s + 2/7·49-s + 3.12·59-s + 0.512·61-s + 0.963·69-s + 0.949·71-s − 1.40·73-s + 0.230·75-s + 1/9·81-s − 1.75·83-s − 1.21·97-s + 0.773·107-s + 1.14·109-s + 0.379·111-s − 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.882887730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882887730\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.441458527347827045314306362484, −9.031290177123584089968069673475, −8.505176949138939806765746980171, −8.173514554803442734671541133944, −7.32328213459775648078965393123, −7.15170496964375224529488571068, −6.75011296795482591362571637167, −5.82998430379572433726607082968, −5.36491761982195510321568888557, −4.75117524706476146238747265245, −4.22069817552665673087881290479, −3.47738164570821613705383822117, −2.73018428216902765717531355398, −2.27617566150396406286489395179, −1.01010663988588321749755795699,
1.01010663988588321749755795699, 2.27617566150396406286489395179, 2.73018428216902765717531355398, 3.47738164570821613705383822117, 4.22069817552665673087881290479, 4.75117524706476146238747265245, 5.36491761982195510321568888557, 5.82998430379572433726607082968, 6.75011296795482591362571637167, 7.15170496964375224529488571068, 7.32328213459775648078965393123, 8.173514554803442734671541133944, 8.505176949138939806765746980171, 9.031290177123584089968069673475, 9.441458527347827045314306362484
