| L(s) = 1 | + 2·3-s + 4·7-s + 9-s − 4·19-s + 8·21-s + 2·25-s − 4·27-s − 16·31-s + 4·37-s + 9·49-s − 8·57-s + 4·63-s + 4·75-s − 11·81-s − 32·93-s + 8·103-s + 28·109-s + 8·111-s − 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 18·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s − 0.917·19-s + 1.74·21-s + 2/5·25-s − 0.769·27-s − 2.87·31-s + 0.657·37-s + 9/7·49-s − 1.05·57-s + 0.503·63-s + 0.461·75-s − 1.22·81-s − 3.31·93-s + 0.788·103-s + 2.68·109-s + 0.759·111-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 1.48·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.935326774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.935326774\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.65974883377888188132622586936, −10.00486810229105835754820765863, −9.272697615700606114690977373546, −8.893121052357834357791015822275, −8.513667798045717088150434618922, −7.919251810222496367542459283074, −7.52936943895418538311873766651, −7.00066723882658864992671123645, −6.04241461459300390013370589718, −5.42737993737237859210571789762, −4.74335651932396086303237226632, −4.05212686002459562297935264367, −3.37831765187502506159651009784, −2.33358272676358500232104327618, −1.72883171187513395133332243413,
1.72883171187513395133332243413, 2.33358272676358500232104327618, 3.37831765187502506159651009784, 4.05212686002459562297935264367, 4.74335651932396086303237226632, 5.42737993737237859210571789762, 6.04241461459300390013370589718, 7.00066723882658864992671123645, 7.52936943895418538311873766651, 7.919251810222496367542459283074, 8.513667798045717088150434618922, 8.893121052357834357791015822275, 9.272697615700606114690977373546, 10.00486810229105835754820765863, 10.65974883377888188132622586936
