| L(s) = 1 | − 2-s + 4-s − 8-s + 2·9-s − 4·13-s + 16-s − 2·17-s − 2·18-s + 12·19-s + 6·25-s + 4·26-s − 32-s + 2·34-s + 2·36-s − 12·38-s + 12·43-s − 2·49-s − 6·50-s − 4·52-s − 12·53-s − 4·59-s + 64-s − 4·67-s − 2·68-s − 2·72-s + 12·76-s − 5·81-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 2/3·9-s − 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 2.75·19-s + 6/5·25-s + 0.784·26-s − 0.176·32-s + 0.342·34-s + 1/3·36-s − 1.94·38-s + 1.82·43-s − 2/7·49-s − 0.848·50-s − 0.554·52-s − 1.64·53-s − 0.520·59-s + 1/8·64-s − 0.488·67-s − 0.242·68-s − 0.235·72-s + 1.37·76-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8678391882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8678391882\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.77907108881444087273641529581, −10.43682675515706924314814057325, −9.585308892003894352620928617008, −9.474930880925621264048140164387, −9.028628480038706929498570955956, −7.963519700774261941083357435383, −7.72312296121947199176988678763, −7.08423628705585568943283930143, −6.70078342341789630626820753769, −5.72630541674926972628712842104, −5.09469118852289810070656842030, −4.45079341164951714924348275703, −3.31769065637062451403331047969, −2.60184502616179409332049081558, −1.25457891793971511405673703129,
1.25457891793971511405673703129, 2.60184502616179409332049081558, 3.31769065637062451403331047969, 4.45079341164951714924348275703, 5.09469118852289810070656842030, 5.72630541674926972628712842104, 6.70078342341789630626820753769, 7.08423628705585568943283930143, 7.72312296121947199176988678763, 7.963519700774261941083357435383, 9.028628480038706929498570955956, 9.474930880925621264048140164387, 9.585308892003894352620928617008, 10.43682675515706924314814057325, 10.77907108881444087273641529581
