L(s) = 1 | − 2·5-s − 2·7-s + 2·9-s − 6·11-s − 6·13-s + 6·19-s + 2·25-s − 12·29-s − 6·31-s + 4·35-s + 6·37-s + 2·41-s + 8·43-s − 4·45-s − 10·47-s + 2·49-s + 12·53-s + 12·55-s − 14·59-s + 28·61-s − 4·63-s + 12·65-s − 10·67-s + 10·71-s + 18·73-s + 12·77-s − 5·81-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.755·7-s + 2/3·9-s − 1.80·11-s − 1.66·13-s + 1.37·19-s + 2/5·25-s − 2.22·29-s − 1.07·31-s + 0.676·35-s + 0.986·37-s + 0.312·41-s + 1.21·43-s − 0.596·45-s − 1.45·47-s + 2/7·49-s + 1.64·53-s + 1.61·55-s − 1.82·59-s + 3.58·61-s − 0.503·63-s + 1.48·65-s − 1.22·67-s + 1.18·71-s + 2.10·73-s + 1.36·77-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7071213228\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7071213228\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−11.43944022927050098461517154926, −10.96173750481516300347425184576, −10.58697352281184241312716992204, −9.905592884860998882103158672412, −9.731584454989350587810646422209, −9.365098295939244004374170985556, −8.758970292404432880813068789228, −7.85093021649198076209079208680, −7.76391438211159487140878230787, −7.22634437943545366626965811223, −7.21131729277581123489759950423, −6.21889772051936640913666958750, −5.58941928845342107136503117268, −5.04094630012188367036566922776, −4.84364698487181914157644063694, −3.67356029911679108549950133231, −3.66367831139067676831793992606, −2.61021113341454942397550605695, −2.18834617534372465151181343882, −0.52370447396557550538899867305,
0.52370447396557550538899867305, 2.18834617534372465151181343882, 2.61021113341454942397550605695, 3.66367831139067676831793992606, 3.67356029911679108549950133231, 4.84364698487181914157644063694, 5.04094630012188367036566922776, 5.58941928845342107136503117268, 6.21889772051936640913666958750, 7.21131729277581123489759950423, 7.22634437943545366626965811223, 7.76391438211159487140878230787, 7.85093021649198076209079208680, 8.758970292404432880813068789228, 9.365098295939244004374170985556, 9.731584454989350587810646422209, 9.905592884860998882103158672412, 10.58697352281184241312716992204, 10.96173750481516300347425184576, 11.43944022927050098461517154926