| L(s) = 1 | − 3-s + 4·7-s − 2·9-s − 8·13-s − 4·16-s − 4·21-s − 9·25-s + 5·27-s + 14·31-s + 6·37-s + 8·39-s + 12·43-s + 4·48-s − 2·49-s − 24·61-s − 8·63-s − 14·67-s − 8·73-s + 9·75-s + 20·79-s + 81-s − 32·91-s − 14·93-s − 14·97-s − 32·103-s − 20·109-s − 6·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s − 2/3·9-s − 2.21·13-s − 16-s − 0.872·21-s − 9/5·25-s + 0.962·27-s + 2.51·31-s + 0.986·37-s + 1.28·39-s + 1.82·43-s + 0.577·48-s − 2/7·49-s − 3.07·61-s − 1.00·63-s − 1.71·67-s − 0.936·73-s + 1.03·75-s + 2.25·79-s + 1/9·81-s − 3.35·91-s − 1.45·93-s − 1.42·97-s − 3.15·103-s − 1.91·109-s − 0.569·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.366222803316740054530803267641, −8.571719581705043558005338128568, −7.927183238362238013907773101180, −7.78789607043645765017922373874, −7.36742097706937993051027630389, −6.42924768192751341652305216268, −6.23870064789214498217427813253, −5.41175071356617719099292784896, −5.01784321266906662288549345024, −4.44574490854809981150981932393, −4.27593882764012714082242634416, −2.65868050490226641918200240944, −2.63044898935838963010520851422, −1.51509492168516879981983515735, 0,
1.51509492168516879981983515735, 2.63044898935838963010520851422, 2.65868050490226641918200240944, 4.27593882764012714082242634416, 4.44574490854809981150981932393, 5.01784321266906662288549345024, 5.41175071356617719099292784896, 6.23870064789214498217427813253, 6.42924768192751341652305216268, 7.36742097706937993051027630389, 7.78789607043645765017922373874, 7.927183238362238013907773101180, 8.571719581705043558005338128568, 9.366222803316740054530803267641
