| L(s) = 1 | − 3-s + 3·5-s + 9-s − 4·13-s − 3·15-s + 4·25-s − 27-s + 13·31-s − 10·37-s + 4·39-s + 2·43-s + 3·45-s + 8·49-s + 12·53-s − 12·65-s + 5·67-s + 9·71-s − 4·75-s − 11·79-s + 81-s + 3·83-s − 15·89-s − 13·93-s + 9·107-s + 10·111-s − 4·117-s + 8·121-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s − 1.10·13-s − 0.774·15-s + 4/5·25-s − 0.192·27-s + 2.33·31-s − 1.64·37-s + 0.640·39-s + 0.304·43-s + 0.447·45-s + 8/7·49-s + 1.64·53-s − 1.48·65-s + 0.610·67-s + 1.06·71-s − 0.461·75-s − 1.23·79-s + 1/9·81-s + 0.329·83-s − 1.58·89-s − 1.34·93-s + 0.870·107-s + 0.949·111-s − 0.369·117-s + 8/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.896888667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.896888667\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
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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
−8.476211313227754476206724395797, −8.174218309025000476387493968734, −7.38238191362612379340214448641, −7.04722735920901527749828115248, −6.67000698613151703063575008992, −6.16090696578192424364111448760, −5.60052280545319940185858041596, −5.41329833483186169241856340238, −4.77453901160876435970590460795, −4.38985193271454291628681175901, −3.65372377716263042164091613036, −2.78029187586195925246544959216, −2.38515385708663421051813528529, −1.69610618856832811159640828965, −0.75864841524264920213428167825,
0.75864841524264920213428167825, 1.69610618856832811159640828965, 2.38515385708663421051813528529, 2.78029187586195925246544959216, 3.65372377716263042164091613036, 4.38985193271454291628681175901, 4.77453901160876435970590460795, 5.41329833483186169241856340238, 5.60052280545319940185858041596, 6.16090696578192424364111448760, 6.67000698613151703063575008992, 7.04722735920901527749828115248, 7.38238191362612379340214448641, 8.174218309025000476387493968734, 8.476211313227754476206724395797
