| L(s) = 1 | − 3·3-s + 4·7-s + 6·9-s − 6·19-s − 12·21-s − 9·27-s + 3·31-s − 37-s + 26·43-s + 9·49-s + 18·57-s + 15·61-s + 24·63-s + 16·67-s − 27·73-s + 13·79-s + 9·81-s − 9·93-s + 33·103-s + 2·109-s + 3·111-s − 11·121-s + 127-s − 78·129-s + 131-s − 24·133-s + 137-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.51·7-s + 2·9-s − 1.37·19-s − 2.61·21-s − 1.73·27-s + 0.538·31-s − 0.164·37-s + 3.96·43-s + 9/7·49-s + 2.38·57-s + 1.92·61-s + 3.02·63-s + 1.95·67-s − 3.16·73-s + 1.46·79-s + 81-s − 0.933·93-s + 3.25·103-s + 0.191·109-s + 0.284·111-s − 121-s + 0.0887·127-s − 6.86·129-s + 0.0873·131-s − 2.08·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.632734744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632734744\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.243487826523598058905383169705, −8.786153947070021355750039809157, −8.668899675287672089802939023410, −7.997227417527295131417847081265, −7.58962573124906349133260680891, −7.48464045462097554541141435737, −6.83972005759554605461724286703, −6.45702835026315145294990548188, −6.13518142396848928272515074892, −5.69616787640139480810198475648, −5.23401023591918113429761458280, −5.10209548304628542923316006719, −4.33014817758926294166073565447, −4.26911537494912470828863027768, −3.92834276564558943061618955119, −2.89955127766283862496053472818, −2.23434729809398625502109834198, −1.87523522277268055875288820052, −1.03775136077836878369368210663, −0.62294794295915339493503069249,
0.62294794295915339493503069249, 1.03775136077836878369368210663, 1.87523522277268055875288820052, 2.23434729809398625502109834198, 2.89955127766283862496053472818, 3.92834276564558943061618955119, 4.26911537494912470828863027768, 4.33014817758926294166073565447, 5.10209548304628542923316006719, 5.23401023591918113429761458280, 5.69616787640139480810198475648, 6.13518142396848928272515074892, 6.45702835026315145294990548188, 6.83972005759554605461724286703, 7.48464045462097554541141435737, 7.58962573124906349133260680891, 7.997227417527295131417847081265, 8.668899675287672089802939023410, 8.786153947070021355750039809157, 9.243487826523598058905383169705
