L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 4·13-s + 5·16-s − 17-s − 4·18-s − 8·19-s − 10·25-s + 8·26-s + 6·32-s − 2·34-s − 6·36-s − 16·38-s + 16·43-s + 2·49-s − 20·50-s + 12·52-s − 12·53-s + 7·64-s + 16·67-s − 3·68-s − 8·72-s − 24·76-s − 5·81-s + 32·86-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2/3·9-s + 1.10·13-s + 5/4·16-s − 0.242·17-s − 0.942·18-s − 1.83·19-s − 2·25-s + 1.56·26-s + 1.06·32-s − 0.342·34-s − 36-s − 2.59·38-s + 2.43·43-s + 2/7·49-s − 2.82·50-s + 1.66·52-s − 1.64·53-s + 7/8·64-s + 1.95·67-s − 0.363·68-s − 0.942·72-s − 2.75·76-s − 5/9·81-s + 3.45·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.450942393\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.450942393\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( 1 + T \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.88013907095950671991603611029, −10.84894668716717132123408330664, −9.942505353869395091050894698746, −9.286071988264369821451329617999, −8.551880873534616105011482646500, −8.107939322961323795970351474484, −7.46682025356819819578371193029, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −5.93029628134094737560230707913, −4.99927116218852259459902565637, −4.05951678114838271873159817135, −3.90229547122613769308947697962, −2.77156487822388402999564657844, −1.95202314698375464079211503776,
1.95202314698375464079211503776, 2.77156487822388402999564657844, 3.90229547122613769308947697962, 4.05951678114838271873159817135, 4.99927116218852259459902565637, 5.93029628134094737560230707913, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 7.46682025356819819578371193029, 8.107939322961323795970351474484, 8.551880873534616105011482646500, 9.286071988264369821451329617999, 9.942505353869395091050894698746, 10.84894668716717132123408330664, 10.88013907095950671991603611029