| L(s) = 1 | + 2·2-s + 2·3-s − 4-s + 4·6-s − 8·8-s + 9-s − 2·12-s − 7·16-s + 12·17-s + 2·18-s − 16·24-s + 2·25-s − 4·27-s − 4·29-s − 4·31-s + 14·32-s + 24·34-s − 36-s + 16·37-s + 12·41-s − 14·48-s + 12·49-s + 4·50-s + 24·51-s − 8·54-s − 8·58-s − 8·62-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s − 1/2·4-s + 1.63·6-s − 2.82·8-s + 1/3·9-s − 0.577·12-s − 7/4·16-s + 2.91·17-s + 0.471·18-s − 3.26·24-s + 2/5·25-s − 0.769·27-s − 0.742·29-s − 0.718·31-s + 2.47·32-s + 4.11·34-s − 1/6·36-s + 2.63·37-s + 1.87·41-s − 2.02·48-s + 12/7·49-s + 0.565·50-s + 3.36·51-s − 1.08·54-s − 1.05·58-s − 1.01·62-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
| \(L(1)\) |
\(\approx\) |
\(3.286731521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.286731521\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.78414043132285524835293765284, −11.50984293883976095861022018174, −10.79589726052525726418589525392, −9.972218133832057026641848592663, −9.772160456705791634158214088772, −9.197067252182714255624063700782, −9.126229915181699995694457107083, −8.340545531112587457453738365807, −7.906301790164895018535236383454, −7.68038036095279809615131372281, −6.88337038976812442964317632653, −5.85021419309098853878634129189, −5.65494328557793259647226186035, −5.47576019278977196294754544847, −4.45874484699166315930893686803, −4.05884877336502081152063320329, −3.64712708925016030135025597587, −2.90934314178071204232927351588, −2.70155918482687699317677189325, −1.03432394308920403536212077347,
1.03432394308920403536212077347, 2.70155918482687699317677189325, 2.90934314178071204232927351588, 3.64712708925016030135025597587, 4.05884877336502081152063320329, 4.45874484699166315930893686803, 5.47576019278977196294754544847, 5.65494328557793259647226186035, 5.85021419309098853878634129189, 6.88337038976812442964317632653, 7.68038036095279809615131372281, 7.906301790164895018535236383454, 8.340545531112587457453738365807, 9.126229915181699995694457107083, 9.197067252182714255624063700782, 9.772160456705791634158214088772, 9.972218133832057026641848592663, 10.79589726052525726418589525392, 11.50984293883976095861022018174, 11.78414043132285524835293765284
