| L(s) = 1 | − 4-s − 2·7-s + 2·13-s − 3·16-s − 8·19-s + 2·28-s − 14·31-s + 4·37-s + 4·43-s − 11·49-s − 2·52-s − 2·61-s + 7·64-s + 10·67-s + 4·73-s + 8·76-s − 20·79-s − 4·91-s − 20·97-s − 32·103-s + 16·109-s + 6·112-s − 19·121-s + 14·124-s + 127-s + 131-s + 16·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s + 0.554·13-s − 3/4·16-s − 1.83·19-s + 0.377·28-s − 2.51·31-s + 0.657·37-s + 0.609·43-s − 1.57·49-s − 0.277·52-s − 0.256·61-s + 7/8·64-s + 1.22·67-s + 0.468·73-s + 0.917·76-s − 2.25·79-s − 0.419·91-s − 2.03·97-s − 3.15·103-s + 1.53·109-s + 0.566·112-s − 1.72·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 139 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.463506946328816080226006467074, −8.425122597355889669889994512870, −7.87274237398111474276796275013, −7.40973264161248255291724076224, −7.01944303036983966171437662303, −6.57151926318146579978012351102, −6.40254105925839265405392752314, −5.96108821827762660836277083165, −5.32768022244652613836548836769, −5.30304421165877357464946860657, −4.49856450083760616897925242466, −4.15790188699369437274084851604, −3.93628002859671980269517249471, −3.41484059990193456283649319907, −2.86218928888147774304922075029, −2.34595394598549049714298547004, −1.84840517166957909244928706288, −1.22063776207964357074678035520, 0, 0,
1.22063776207964357074678035520, 1.84840517166957909244928706288, 2.34595394598549049714298547004, 2.86218928888147774304922075029, 3.41484059990193456283649319907, 3.93628002859671980269517249471, 4.15790188699369437274084851604, 4.49856450083760616897925242466, 5.30304421165877357464946860657, 5.32768022244652613836548836769, 5.96108821827762660836277083165, 6.40254105925839265405392752314, 6.57151926318146579978012351102, 7.01944303036983966171437662303, 7.40973264161248255291724076224, 7.87274237398111474276796275013, 8.425122597355889669889994512870, 8.463506946328816080226006467074
