| L(s) = 1 | + 4·3-s − 2·5-s + 4·7-s + 8·9-s + 6·13-s − 8·15-s − 6·17-s + 16·21-s − 12·23-s − 25-s + 12·27-s − 8·35-s + 6·37-s + 24·39-s + 12·43-s − 16·45-s − 12·47-s + 8·49-s − 24·51-s + 6·53-s + 16·59-s + 12·61-s + 32·63-s − 12·65-s − 12·67-s − 48·69-s + 10·73-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 1.51·7-s + 8/3·9-s + 1.66·13-s − 2.06·15-s − 1.45·17-s + 3.49·21-s − 2.50·23-s − 1/5·25-s + 2.30·27-s − 1.35·35-s + 0.986·37-s + 3.84·39-s + 1.82·43-s − 2.38·45-s − 1.75·47-s + 8/7·49-s − 3.36·51-s + 0.824·53-s + 2.08·59-s + 1.53·61-s + 4.03·63-s − 1.48·65-s − 1.46·67-s − 5.77·69-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.508260149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.508260149\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−9.497023981196319156069442388037, −9.472594542764465351869422806920, −8.675369561160211760239252900619, −8.586260181435372710200430713267, −8.300573285801637366777779259759, −8.091392288197894233182200964479, −7.65420646679991861856259195737, −7.35598245171220023068329107050, −6.74649232761363064392930011958, −6.11727617828772137609622625213, −5.85051682060622969539971059111, −5.00337298989382320448059874226, −4.30965528586062053402766318334, −4.26364627071486461401030005654, −3.68991080273300188380863296490, −3.47116596291589260258044269599, −2.59397070103914238931946187609, −2.08154550230799819570622829109, −1.91034882812787401974425378821, −0.877181107635094187127732591570,
0.877181107635094187127732591570, 1.91034882812787401974425378821, 2.08154550230799819570622829109, 2.59397070103914238931946187609, 3.47116596291589260258044269599, 3.68991080273300188380863296490, 4.26364627071486461401030005654, 4.30965528586062053402766318334, 5.00337298989382320448059874226, 5.85051682060622969539971059111, 6.11727617828772137609622625213, 6.74649232761363064392930011958, 7.35598245171220023068329107050, 7.65420646679991861856259195737, 8.091392288197894233182200964479, 8.300573285801637366777779259759, 8.586260181435372710200430713267, 8.675369561160211760239252900619, 9.472594542764465351869422806920, 9.497023981196319156069442388037
