| L(s) = 1 | − 4-s + 16-s + 4·25-s − 2·37-s − 2·43-s − 2·64-s + 2·67-s + 2·79-s − 4·100-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s + 4·25-s − 2·37-s − 2·43-s − 2·64-s + 2·67-s + 2·79-s − 4·100-s + 4·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 2·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.353911955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353911955\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.22526766865111405430422014758, −5.92008926471537642011629601388, −5.72088573612729399074510549131, −5.35975289850703365644057180774, −5.31606543546773728417212005199, −5.20550860413768357451081326286, −5.03973262411922178497468603429, −4.64613576053812417333478091661, −4.60022366455408694744054172487, −4.59785516865929703006415627313, −4.28138057149627682053376202321, −3.82821516926686114720949338764, −3.68847735340651066749710323682, −3.54191110121503720780547574156, −3.26124633349112186740704832729, −3.11552220910246532381051261252, −2.91401199281911393226795243872, −2.81592891573802970161485433766, −2.17209944814564299118856601607, −2.11559349590874141738305129526, −1.87321597079727962376842263858, −1.49302199674201314704712759050, −1.06746193244147675682369422061, −0.961905073134386469154891176553, −0.50854380235820461590738943848,
0.50854380235820461590738943848, 0.961905073134386469154891176553, 1.06746193244147675682369422061, 1.49302199674201314704712759050, 1.87321597079727962376842263858, 2.11559349590874141738305129526, 2.17209944814564299118856601607, 2.81592891573802970161485433766, 2.91401199281911393226795243872, 3.11552220910246532381051261252, 3.26124633349112186740704832729, 3.54191110121503720780547574156, 3.68847735340651066749710323682, 3.82821516926686114720949338764, 4.28138057149627682053376202321, 4.59785516865929703006415627313, 4.60022366455408694744054172487, 4.64613576053812417333478091661, 5.03973262411922178497468603429, 5.20550860413768357451081326286, 5.31606543546773728417212005199, 5.35975289850703365644057180774, 5.72088573612729399074510549131, 5.92008926471537642011629601388, 6.22526766865111405430422014758
