L(s) = 1 | + 2·2-s − 4-s − 8·8-s + 6·9-s − 4·13-s − 7·16-s + 12·18-s + 8·19-s + 6·25-s − 8·26-s + 14·32-s − 6·36-s + 16·38-s − 8·43-s − 2·49-s + 12·50-s + 4·52-s − 12·53-s + 24·59-s + 35·64-s + 8·67-s − 48·72-s − 8·76-s + 27·81-s + 8·83-s − 16·86-s + 20·89-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s + 2·9-s − 1.10·13-s − 7/4·16-s + 2.82·18-s + 1.83·19-s + 6/5·25-s − 1.56·26-s + 2.47·32-s − 36-s + 2.59·38-s − 1.21·43-s − 2/7·49-s + 1.69·50-s + 0.554·52-s − 1.64·53-s + 3.12·59-s + 35/8·64-s + 0.977·67-s − 5.65·72-s − 0.917·76-s + 3·81-s + 0.878·83-s − 1.72·86-s + 2.11·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.252664248\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.252664248\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 17 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−12.11576759312387413270668491299, −11.96576197067687149047187743735, −11.38819580280739913467497946584, −10.55060498663064660633869541862, −9.895173764373709135343902008222, −9.785764555801647966760879556209, −9.404339157686116863184330181939, −8.822103400278922359567134491236, −8.151160348286873343769939957815, −7.62628132785994960680224184756, −6.93489966522597287012637266133, −6.67831097603677793979582836253, −5.82753729752985931996525656458, −5.04654405096225976917833244370, −4.96451134545497643237637350602, −4.53772610926885161211825836280, −3.61419840993050833045566105447, −3.48509309417351538944822117777, −2.44234076885814509524119187057, −1.01140502692843763436254945597,
1.01140502692843763436254945597, 2.44234076885814509524119187057, 3.48509309417351538944822117777, 3.61419840993050833045566105447, 4.53772610926885161211825836280, 4.96451134545497643237637350602, 5.04654405096225976917833244370, 5.82753729752985931996525656458, 6.67831097603677793979582836253, 6.93489966522597287012637266133, 7.62628132785994960680224184756, 8.151160348286873343769939957815, 8.822103400278922359567134491236, 9.404339157686116863184330181939, 9.785764555801647966760879556209, 9.895173764373709135343902008222, 10.55060498663064660633869541862, 11.38819580280739913467497946584, 11.96576197067687149047187743735, 12.11576759312387413270668491299