L(s) = 1 | − 4·2-s − 3-s + 8·4-s + 4·6-s − 8·8-s − 2·9-s + 11-s − 8·12-s − 4·16-s − 4·17-s + 8·18-s − 4·22-s + 8·24-s − 9·25-s + 5·27-s + 14·31-s + 32·32-s − 33-s + 16·34-s − 16·36-s + 6·37-s − 16·41-s + 8·44-s + 4·48-s − 10·49-s + 36·50-s + 4·51-s + ⋯ |
L(s) = 1 | − 2.82·2-s − 0.577·3-s + 4·4-s + 1.63·6-s − 2.82·8-s − 2/3·9-s + 0.301·11-s − 2.30·12-s − 16-s − 0.970·17-s + 1.88·18-s − 0.852·22-s + 1.63·24-s − 9/5·25-s + 0.962·27-s + 2.51·31-s + 5.65·32-s − 0.174·33-s + 2.74·34-s − 8/3·36-s + 0.986·37-s − 2.49·41-s + 1.20·44-s + 0.577·48-s − 1.42·49-s + 5.09·50-s + 0.560·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11979 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.90721695501259097313070074829, −10.08710379123756686578115682884, −10.03550909718107888433464868208, −9.461316816361475492652505870168, −8.603539619290756001226038948684, −8.571719581705043558005338128568, −7.927183238362238013907773101180, −7.36742097706937993051027630389, −6.42924768192751341652305216268, −6.36261389471308870138602900888, −5.01784321266906662288549345024, −4.27593882764012714082242634416, −2.65868050490226641918200240944, −1.51509492168516879981983515735, 0,
1.51509492168516879981983515735, 2.65868050490226641918200240944, 4.27593882764012714082242634416, 5.01784321266906662288549345024, 6.36261389471308870138602900888, 6.42924768192751341652305216268, 7.36742097706937993051027630389, 7.927183238362238013907773101180, 8.571719581705043558005338128568, 8.603539619290756001226038948684, 9.461316816361475492652505870168, 10.03550909718107888433464868208, 10.08710379123756686578115682884, 10.90721695501259097313070074829