| L(s) = 1 | − 4-s + 4·13-s + 16-s − 6·17-s + 12·23-s + 25-s − 2·43-s + 5·49-s − 4·52-s + 12·53-s − 16·61-s − 64-s + 6·68-s + 20·79-s − 12·92-s − 100-s − 24·101-s + 28·103-s + 24·107-s + 12·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 2.50·23-s + 1/5·25-s − 0.304·43-s + 5/7·49-s − 0.554·52-s + 1.64·53-s − 2.04·61-s − 1/8·64-s + 0.727·68-s + 2.25·79-s − 1.25·92-s − 0.0999·100-s − 2.38·101-s + 2.75·103-s + 2.32·107-s + 1.12·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261856548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261856548\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−12.45294124861700791440249131306, −11.98498683772879958086154389649, −11.23684790378403051651080020099, −11.01456061726473452845535742694, −10.66614267660499091361344189900, −10.03187854458693744747090164165, −9.326788776893706325905715770669, −8.908681650731814745036786789141, −8.717010246469857281133556086927, −8.155078257063527163934409199324, −7.17259931635885781992660017651, −7.11599848403789371237621767225, −6.23338042069986682306780114159, −5.86020965033864823584977628847, −4.80868965195490964461641557645, −4.78537988733860045503255159505, −3.78249240269048129685069889240, −3.21574836952841659401558933459, −2.26604358136427699351263497580, −1.01420646281854320776568542947,
1.01420646281854320776568542947, 2.26604358136427699351263497580, 3.21574836952841659401558933459, 3.78249240269048129685069889240, 4.78537988733860045503255159505, 4.80868965195490964461641557645, 5.86020965033864823584977628847, 6.23338042069986682306780114159, 7.11599848403789371237621767225, 7.17259931635885781992660017651, 8.155078257063527163934409199324, 8.717010246469857281133556086927, 8.908681650731814745036786789141, 9.326788776893706325905715770669, 10.03187854458693744747090164165, 10.66614267660499091361344189900, 11.01456061726473452845535742694, 11.23684790378403051651080020099, 11.98498683772879958086154389649, 12.45294124861700791440249131306
