| L(s) = 1 | − 4·5-s + 6·11-s − 4·16-s + 11·25-s − 10·29-s − 4·31-s + 4·41-s − 24·55-s − 20·59-s + 16·61-s + 16·71-s + 10·79-s + 16·80-s + 24·101-s − 10·109-s + 5·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + 151-s + 16·155-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.80·11-s − 16-s + 11/5·25-s − 1.85·29-s − 0.718·31-s + 0.624·41-s − 3.23·55-s − 2.60·59-s + 2.04·61-s + 1.89·71-s + 1.12·79-s + 1.78·80-s + 2.38·101-s − 0.957·109-s + 5/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 1.28·155-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.113187250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.113187250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.185610466549716491373020878880, −8.914934077267715540340110292735, −8.477273678864461927943954032713, −8.068685431207513119463679726820, −7.60266642484173485727207332162, −7.43150220908455889732865608073, −6.86755711341955704127462141106, −6.71667020116760375332818104541, −6.25198238806270656338770498791, −5.70423934370409127827477889760, −5.22465016424296750935251716651, −4.50249991809603940604306611287, −4.50225760153827742349036908171, −3.84792394606019716450782592531, −3.55407865366965380939059433745, −3.35402035934725302253072320882, −2.41005433314244717753056632276, −1.91245704080113068690707135803, −1.15704039761806755584204998752, −0.41304973719446752448514052961,
0.41304973719446752448514052961, 1.15704039761806755584204998752, 1.91245704080113068690707135803, 2.41005433314244717753056632276, 3.35402035934725302253072320882, 3.55407865366965380939059433745, 3.84792394606019716450782592531, 4.50225760153827742349036908171, 4.50249991809603940604306611287, 5.22465016424296750935251716651, 5.70423934370409127827477889760, 6.25198238806270656338770498791, 6.71667020116760375332818104541, 6.86755711341955704127462141106, 7.43150220908455889732865608073, 7.60266642484173485727207332162, 8.068685431207513119463679726820, 8.477273678864461927943954032713, 8.914934077267715540340110292735, 9.185610466549716491373020878880
